On the geometry of the countably branching diamond graphs
Florent P. Baudier, Ryan Causey, Stephen DIlworth, Denka Kutzarova, Nirina L. Randrianarivony, Thomas Schlumprecht, Sheng Zhang
aa r X i v : . [ m a t h . M G ] D ec ON THE GEOMETRY OF THE COUNTABLY BRANCHINGDIAMOND GRAPHS
FLORENT BAUDIER, RYAN CAUSEY, STEPHEN DILWORTH, DENKA KUTZAROVA,N. L. RANDRIANARIVONY, THOMAS SCHLUMPRECHT, AND SHENG ZHANG
Abstract.
In this article, the bi-Lipschitz embeddability of the sequence ofcountably branching diamond graphs ( D ωk ) k ∈ N is investigated. In particular itis shown that for every ε > k ∈ N , D ωk embeds bi-Lipschiztly with distor-tion at most 6(1+ ε ) into any reflexive Banach space with an unconditional as-ymptotic structure that does not admit an equivalent asymptotically uniformlyconvex norm. On the other hand it is shown that the sequence ( D ωk ) k ∈ N doesnot admit an equi-bi-Lipschitz embedding into any Banach space that has anequivalent asymptotically midpoint uniformly convex norm. Combining thesetwo results one obtains a metric characterization in terms of graph preclusionof the class of asymptotically uniformly convexifiable spaces, within the classof separable reflexive Banach spaces with an unconditional asymptotic struc-ture. Applications to bi-Lipschitz embeddability into L p -spaces and to someproblems in renorming theory are also discussed. Contents
1. Introduction 21.1. Motivation 21.2. Content of the article 42. The countably branching diamond graphs 62.1. Graph theoretical recursive definition 72.2. Non recursive definition 72.3. Basic metric properties of the countably branching diamond graphs 113. Embeddability of the countably branching diamond graphs 123.1. Embeddability into Banach spaces containing particular ℓ ∞ -trees 123.1.1. Good ℓ ∞ -trees of arbitrary height 133.1.2. The embedding 133.2. Banach spaces containing good ℓ ∞ -trees of arbitrary height 193.2.1. Asymptotic properties and trees 193.2.2. Sufficient conditions for the containment of good ℓ ∞ -trees 213.3. Embeddability into L L p ( Y ) 314. Non-embeddability of the countably branching diamond graphs 355. Applications 40References 42 Mathematics Subject Classification. Introduction
Motivation.
One of the most natural way to grasp the geometry of a metricspace is to understand in which metric spaces, in particular which Banach spaces,it does, or it does not, bi-Lipschitzly embed. In this article the geometry of thecountably branching diamond graph of depth k , denoted D ωk , is studied. In thisintroduction only a few fundamental notions and concept from metric space andBanach space geometry are recalled and the reader is directed to the core of thearticle or the references [13], [36] for undefined notions.Let ( X, d X ) and ( Y, d Y ) be two metric spaces. A map f : X → Y is called abi-Lipschitz embedding if there exist s > D ≥ x, y ∈ X ,(1) s · d X ( x, y ) ≤ d Y ( f ( x ) , f ( y )) ≤ D · s · d X ( x, y ) . As usual c Y ( X ) := inf { D ≥ | equation (1) holds for some embedding f } denotesthe Y -distortion of X . If there is no bi-Lipschitz embedding from X into Y then weset c Y ( X ) = ∞ . A sequence ( X k ) ∈ N of metric spaces is said to equi-bi-Lipschitzlyembed into a metric space Y if sup k ∈ N c Y ( X k ) < ∞ .The research carried out in this article is motivated by exhibiting analogies ordiscrepancies between certain metric characterizations of local properties of Banachspaces and their asymptotic counterparts. To make this more precise some usefulterminology needs to be introduced. In general, one is seeking metric characteriza-tions in terms of metric space preclusion in a nonlinear category. For concreteness,let us state one of the numerous metaproblems that can be considered. Say, onewants to work in the uniform category where the objects are metric spaces andthe morphisms are injective uniformly continous maps. Let P be a property ofmetric spaces or Banach spaces, and C P the class it canonically defines. An inter-esting problem, is to find a family of metric spaces ( X i ) i ∈ I such that Y ∈ C P ifand only if the family ( X i ) i ∈ I does not equi-uniformly embed into Y . Arguably,the most interesting case belongs to the Lipschitz category and deals with Banachspace properties. The following metaproblem is general enough to encompass allthe known metric characterizations in terms of metric space preclusion. Problem 1.1 (Metric space preclusion characterizations) . Fix an ambient classof Banach spaces B amb . Are there Banach space properties P (or equivalently theclasses they define B P ), and families of metric spaces ( X i ) i ∈ I such that if Y ∈ B amb ,then(1) if Y ∈ B P then sup i ∈ I c Y ( X i ) = ∞ ,(2) if Y / ∈ B P then sup i ∈ I c Y ( X i ) < ∞ .Following [34] , the family ( X i ) i ∈ I is called in that case a family of test-spaces for P (or B P ) within the class B amb in the Lipschitz category.If moreover,(2’) sup Y / ∈B P sup i ∈ I c Y ( X i ) < ∞ ,the family ( X i ) i ∈ I shall be called a uniformly characterizing family for P (or B P )within the class B amb in the Lipschitz category. Since we will only deal with the Lipschitz category in this article we will fromnow on drop the mention to the Lipschitz category. We are mainly concernedwith sequences of test-spaces in this article and if the ambient class is the classof all Banach spaces one shall simply say “sequence of test-spaces” or “uniformlycharacterizing sequence”. Note that if ( X k ) k ∈ N is a sequence of test-spaces for B P then Y ∈ B P ⇐⇒ sup k ∈ N c Y ( X k ) = ∞ , or equivalently, Y / ∈ B P ⇐⇒ sup k ∈ N c Y ( X k ) < ∞ . Remark that whenever B P admits a sequence of test-spaces N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 3 then B P is stable under bi-Lipschitz embeddability, in particular P must be heredi-tary (i.e. passes to subspaces). Also, when sup k ∈ N c Y ( X k ) = ∞ , estimating the rateof growth of c Y ( X k ) in terms of some numerical parameter that can be associatedto B P and the sequence can be fundamental for applications.Ribe’s rigidity theorem [41] suggests that it is reasonable to believe that localproperties of Banach spaces could be characterized in purely metric terms (notnecessarily in terms of test-spaces though). The first successful step in the Ribeprogram was obtained by Bourgain when he showed that the sequence ( B k ) k ∈ N ofbinary trees of height k is a uniformly characterizing sequence for super-reflexivity.We refer to [2] and [31] for a thorough description of the Ribe program and its suc-cessful achievements. It is worth noticing at this point that an analogue of Ribe’srigidity theorem in the asymptotic setting remains elusive. Nevertheless, a viableanalogue of Bourgain’s super-reflexivity characterization in the asymptotic settingwas obtained in [4]. Before we state the Baudier-Kalton-Lancien characterization,let us mention that when dealing with asymptotic properties one shall restrict one-self to the class of reflexive Banach spaces. Whether reflexivity is a technical ora conceptual requirement seems to be a challenging and delicate issue. Also, theBaudier-Kalton-Lancien characterization could be considered as the first step in acertain asymptotic declination of the Ribe program which would seek for charac-terizations of asymptotic properties in purely metric terms. Important techniques,deep contributions, and profound ideas to tackle such a program are ubiquitous inrecent work of the late Nigel Kalton (and his coauthors; c.f. [17], [18], [4], [19], [20],[21], [22]).Recall that a Banach space Y is said to be uniformly convexifiable if Y admits an equivalent norm that is uniformly convex, and the convenient notation Y ∈ h U C i shall be used in the sequel. Similar notations and acronyms are used without fur-ther explanations for renorming with other properties. The Baudier-Kalton-Lanciencharacterization states that the sequence ( T ωk ) k ∈ N of countably branching trees ofheight k is a uniformly characterizing sequence for the class of Banach spaces thatare asymptotically uniformly smoothable and asymptotically uniformly convexi-fiable, within the class of reflexive Banach spaces. The paramount ingredient toachieve this characterization is the Szlenk index. Indeed in the reflexive setting, theBanach spaces that are asymptotically uniformly smoothable and asymptoticallyuniformly convexifiable are exactly those whose Szlenk index and the Szlenk indexof their dual is at most ω . Around the same time, Johnson and Schechtman [15]found two new uniformly characterizing sequences for super-reflexivity, namely thesequence ( D k ) k ∈ N of (2-branching) diamond graphs and the sequence ( L k ) k ∈ N of(2-branching) Laakso graphs. Very recently, Ostrovskii and Randrianantoanina [37]showed that for any r ≥ D rk ) k ∈ N of r -branching diamond graphs isalso a uniformly characterizing sequence for super-reflexivity. In [5], Baudier andZhang gave a different proof that sup k ∈ N c Y ( T ωk ) = ∞ where Y is any reflexiveBanach space that is asymptotically uniformly smoothable and asymptotically uni-formly convexifiable. This new proof is based on the fact that such spaces have anequivalent norm that has property ( β ) of Rolewicz (see [10]) and could be easilyadjusted, for the same Banach space target, to show that sup k ∈ N c Y ( L ωk ) = ∞ andsup k ∈ N c Y ( P ωk ) = ∞ where ( L ωk ) k ∈ N (resp. ( P ωk ) k ∈ N ) is the sequence of countablybranching Laakso (resp. parasol) graphs. Unfortunately the geometric argumentused in this proof could not settle the similar problem for the sequence of count-ably branching diamond graphs. This issue is resolved in this article. Also, aquestion that arises naturally is whether one could replace the sequence ( T ωk ) k ∈ N inthe Baudier-Kalton-Lancien characterization by ( D ωk ) k ∈ N , ( L ωk ) k ∈ N or ( P ωk ) k ∈ N andthus obtain another analogy between the local and the asymptotic versions of the BAUDIER ET AL.
Ribe program. One of the main application of our work is that one cannot replacethe sequence ( T ωk ) k ∈ N neither with the sequence ( D ωk ) k ∈ N , ( L ωk ) k ∈ N nor ( P ωk ) k ∈ N inthe Baudier-Kalton-Lancien characterization. This seemingly discrepancy betweenthe local Ribe program and the asymptotic Ribe program is due to the fact that theclass of super-reflexive Banach spaces, the class of uniformly convexifiable Banachspaces, and the class of uniformly smoothable Banach spaces form the same classin different disguise! However, the classes of uniform asymptotically convexifiableand uniform asymptotically smoothable spaces differ. Actually our work showsthat the sequence ( D ωk ) k ∈ N is a uniformly characterizing sequence for the class ofasymptotically uniformly convexifiable Banach spaces within the class of separablereflexive Banach spaces with an unconditional asymptotic structure.1.2. Content of the article.
In Section 2 two descriptions of the countablybranching diamond graph D ωk are given: a recursive graph-theoretical descriptionand a non-recursive set-theoretical description. The two constructions define thesame unweighted graph (up to graph isomorphism) and therefore the same metricspace (up to isometry). The basic metric properties of the diamond graphs arerecorded in Section 2.3 for convenience.Section 3 is divided into four parts, three of them dealing with the embeddabil-ity of the sequence ( D ωk ) k ∈ N into certain Banach spaces. In Section 3.1 we showthat for every k ∈ N , c Y ( D ωk ) ≤ Y contains arbitrarily good ℓ <ω ∞ -trees.In Section 3.3 a new proof is given showing that for any k ∈ N , D ωk admits abi-Lipschitz embedding into L [0 ,
1] that has distortion at most 2. Note that theembedding possesses some interesting properties, such as being “vertically isomet-ric”. Both embeddings requires the non-recursive set-theoretical description of thecountably branching diamond graphs. In Section 3.4 embeddability into Banachspaces of the form L p ([0 , , Y ) is discussed. It is shown that for 1 ≤ p < ∞ , if Y is not super-reflexive, then for every k ∈ N , c L p ([0 , ,Y ) ( D ωk ) ≤ /p . This resulthas an interesting consequence in the renorming theory that is discussed in Section5. The remaining part, Section 3.2, is devoted to providing sufficient conditionsfor a Banach space to contain arbitrarily good ℓ ∞ -trees of arbitrary height, andthus is essentially Banach space theoretical. In particular we show that such treescan always be found in any Banach space that contains, for all n ∈ N , ℓ n ∞ in its n -th asymptotic structure. Combining this result with the embedding result fromSection 3.1 one obtains our main embedding theorem. Theorem A.
If for all n ∈ N , ℓ n ∞ is in the n -th asymptotic structure of Y , then sup k ∈ N c Y ( D ωk ) < ∞ , i.e. the sequence ( D ωk ) k ∈ N can be equi-bi-Lipschtizly embeddedinto Y . Finally, we show that any separable reflexive Banach space with an unconditionalasymptotic structure whose Szlenk index of its dual is larger than ω contains ℓ n ∞ in its n -th asymptotic structure for all n ∈ N . To achieve this, some tools andconcepts from asymptotic Banach space theory (e.g. trees and branches in Banachspaces, asymptotic structure...) are needed and they are recalled in Section 3.2.1.In Section 4 we are concerned with finding obstructions to the embeddability ofthe diamond graphs. The main result of this section states that one cannot em-bed equi-bi-Lipschitzly the sequence ( D ωk ) k ∈ N into a Banach space that is midpointasymptotically uniformly convex. The proof of the main result builds upon twovery useful techniques: an approximate midpoint argument (originally from En-flo) and the self-improvement argument `a la Johnson and Schechtman. The proofcan be significantly generalized to handle a quite large collection of sequences of N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 5 graphs, which in particular contains the countably branching Laakso graphs andthe countably branching parasol graphs.The article ends with a section devoted to the applications of our work in metricgeometry, and in renorming theory, some of which being already mentioned in thefirst part of this introduction. The two main applications are a new metric char-acterization in terms of graph preclusion, and tight estimates on the L p -distortionof the countably branching diamond graphs. Our asymptotic notation conventionsare the following. Throughout this article we will use the notation . or & , to de-note the corresponding inequalities up to universal constant factors. We will alsodenote equivalence up to universal constant factors by ≈ , i.e., A ≈ B is the sameas ( A . B ) ∧ ( A & B ). Theorem B.
Let Y be a reflexive Banach space with an unconditional asymp-totic structure. Then, Y is asymptotically uniformly convexifiable if and only if sup k ∈ N c Y ( D ωk ) = ∞ . Theorem C. i) For ≤ p < ∞ , c ℓ p ( D ωk ) ≈ k /p .ii) For < p < ∞ , c L p ( D ωk ) ≈ min { k /p , √ k ) } . We conclude this section by gathering all the known (as far as we know) metricspace preclusion characterizations in the following three tables. Note that all theknown sequence of test-spaces are actually uniformly characterizing sequences. Wealso included some interesting open problems. Needless to say that they are manymore open problems. B amb B P test-spacessuper-reflexivity ( B k ) k ∈ N , ( T rk ) k ∈ N r ≥ h UC i B ∞ Baudier ([3],2007)Dz( Y ) = ω ( D k ) k ∈ N , ( L k ) k ∈ N Johnson-Schechtman ([15],2009)all Banach spaces h US i certain hyperbolic groups Ostrovskii ([35],2014)Dz( Y ∗ ) = ω ( D rk ) k ∈ N , r ≥ H k ) k ∈ N Bourgain-Milman-Wolfsonnon-trivial type ([9],1986)non-trivial cotype ℓ ∞ -grids ([ m ] n ∞ ) m,n ∈ N Mendel-Naor ([28], 2008)UMD openPisier’s property ( α ) open Table 1.
Local Ribe program
BAUDIER ET AL. B amb B P test-spaces h AUC i ∩ h
AUS i ( T ωk ) k ∈ N , T ω ∞ Baudier-Kalton-Lancien([4],2010) h ( β ) i reflexive Banach spaces max { Sz( Y ) , Sz( Y ∗ ) } = ω ( S , d , ) Motakis-Schlumprecht([30],2016)max { Sz( Y ) , Sz( Y ∗ ) } ≤ ω α ( S α , d ,α ) Motakis-Schlumprechtwhen ω α = α ([30],2016)reflexive Banach spaces h AUC i with an unconditional ( D ωk ) k ∈ N this articleasymptotic structure Sz( Y ∗ ) = ω h AUS i openSz( Y ) = ω Table 2.
Asymptotic Ribe program B amb B P test-spacesdual Banach spaces Radon-Nikod´ym D ∞ Ostrovskii ([34], 2014)reflexivity open
Table 3.
Extended Ribe program The countably branching diamond graphs
By a graph G = ( V, E ) we mean a graph with vertex-set V and edge-set E . The set of vertices or edges is allowed to be infinite however we restrictour attention to simple graphs. i.e. without multiple edges. Our graphswill always be unweighted and equipped with the shortest path distance.The geometry of the (binary or 2-branching) diamond graphs has provedto be fundamental in connection with applications in theoretical computerscience (for the dimension reduction problem in ℓ see for instance [25]).The diamond graph of depth k is build in a recursive fashion starting withthe 4-cycle (the diamond graph of depth 1). The diamond graph of depth 2is obtained by replacing each edge of the diamond graph of depth 1 with acopy of the 4-cycle. The diamond graphs of higher depth are build in sim-ilar fashion. The sequence of graphs obtained, denoted ( D k ) k ∈ N , is usuallysimply refered to as the sequence of diamond graphs . In this article weare concerned with the sequence of countably branching diamond graphs,denoted ( D ωk ) k ∈ N , whose formal definition is presented in the next section. k + 1 is obtained by replacing each edge of the diamond graph of depth k by acopy of itself; this actually gives a subsequence of the sequence of the diamond graphs as definedhere. N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 7
Graph theoretical recursive definition.
A directed s - t graph G =( V, E ) is a directed graph which has two distinguished vertices s, t ∈ V . Toavoid confusion, we will also write sometimes s ( G ) and t ( G ). There is anatural way to “compose” directed s - t graphs using the ⊘ -product definedin [26]. Given two directed s - t graphs H and G , define a new graph H ⊘ G as follows:i) V ( H ⊘ G ) := V ( H ) ∪ ( E ( H ) × ( V ( G ) \{ s ( G ) , t ( G ) } ))ii) For every oriented edge e = ( u, v ) ∈ E ( H ), there are | E ( G ) | orientededges, (cid:8) ( { e, v } , { e, v } ) | ( v , v ) ∈ E ( G ) and v , v / ∈ { s ( G ) , t ( G ) } (cid:9) ∪ (cid:8) ( u, { e, w } ) | ( s ( G ) , w ) ∈ E ( G ) (cid:9) ∪ (cid:8) ( { e, w } , u ) | ( w, s ( G )) ∈ E ( G ) (cid:9) ∪ (cid:8) ( { e, w } , v ) | ( w, t ( G )) ∈ E ( G ) (cid:9) ∪ (cid:8) ( v, { e, w } ) | ( t ( G ) , w ) ∈ E ( G ) (cid:9) iii) s ( H ⊘ G ) = s ( H ) and t ( H ⊘ G ) = t ( H ).It is also clear that the ⊘ -product is associative (in the sense of graph-isomorphism or metric space isometry), and for a directed graph G one canrecursively define G ⊘ k for all k ∈ N as follows: • G ⊘ := G . • G ⊘ k +1 := G ⊘ k ⊘ G , for k ≥ G ⊘ to be the two-vertex graph with an edge connecting them. Notealso that if the base graph G is symmetric the graph G ⊘ k does not dependon the orientation of the edges.Consider the complete bipartite infinite graph K ,ω with two vertices onone side, (such that one is s ( K ,ω ) and the other t ( K ,ω )), and countablymany vertices on the other side. The countably branching diamond graph ofdepth k is defined as D ωk := K ⊘ k ,ω . If one starts with the complete bipartitegraph K ,r for some r ≥ r -branchingdiamond graph of depth k . In particular D k := K ⊘ k , .The recursive definition of the various types of diamond graphs (and thebasic metric properties that can be derived from it) is usually sufficient toprove metric statement about them. However in this article we will need amore “concrete” representation of the countably branching diamond graphsin order to prove our main embedding result. This representation, or coding,is the purpose of the next section.2.2. Non recursive definition.
We denote by [ N ] all subsets of N , by [ N ] <ω all finite subsets, and by [ N ] ω all infinite subsets. For k ∈ N := N ∪ { } wedenote by [ N ] ≤ k the subsets of N which have at most k elements, by [ N ]
0) and t ω = ( ∅ , V = { b ω , t ω } ∪ (cid:8)(cid:0) { j } , (cid:1) : j ∈ N (cid:9) and E = (cid:8) { b ω , ( { j } , ) } , { t ω , ( { j } , ) } : j ∈ N (cid:9) .D ω b ( ∅ , b ( ∅ , D ω b ( ∅ , b b b b b b b b b ( { } , ) ( { j } , ) b ( ∅ , D ω b b bbbbb b bb b bbbbb b bb bb bbb bb bb b b b b bb bbbb b b bb b b bb b bb b b ( ∅ , { , } , )( { } , ) ( { j } , )( { , } , ) ( ∅ , In the following lemma we gather the basic combinatorial properties ofthe sequence ( G k ) k ∈ N that will be needed in the sequel. Elements of proofsare only given for the facts that are not completely obvious. N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 9
Lemma 2.1. a) For j, k ∈ N , with j ≤ k it follows that V j ⊂ V k .b) Assume that k ∈ N and ( A, r ) , ( B, s ) are in V k with (cid:8) ( A, r ) , ( B, s ) (cid:9) ∈ E k .Then it follows that | r − s | = 2 − k and either | A | = k and | B | < k or | B | = k and | A | < k .c) Let k ∈ N . If ( B, s ) ∈ V k , with B ∈ [ N ] k (and thus s ∈ B k ), and if ( A, r ) ∈ V k , then it follows that { ( A, r ) , ( B, s ) } ∈ E k if and only if (2) A ≺ B and (3) either r = s − − k = i − X i =1 σ i − i , and A = B | i − ,if i − = max { ≤ i ≤ k − σ i = 1 } exists, and A = ∅ and r = 0 otherwise , or r = s + 2 − k = i + − X i =1 σ i − i + 2 − i + , and A = B | i + if i + = max { ≤ i ≤ k − σ i = 0 } exists, and A = ∅ and r = 1 otherwise . d) For a given ( B, r ) ∈ [ N ] k × B k there is a unique edge (cid:8) ( B + , r +2 − k ) , ( B − , r − − k ) } ∈ E k − , for which we have { ( B, r ) , ( B + , r + 2 − k ) } ∈ E k and { ( B, r ) , ( B − , r − − k ) } ∈ E k . Proof. c) Assume k ≥
1. From the definition of E k it follows that iffor two elements u and v in V k , say u = ( A, r ) and v = ( B, s ),we have { u, v } ∈ E k , then either A or B has cardinality k and theother set does have cardinality less than k . So assume that | A | < | B | = k . Then write s as s = P ki =1 σ i − i , with σ i ∈ { , } k , for i ∈ { , , . . . , k } and σ k = 1. Now since | r − s | = 2 − k and r ∈ B | A | itfollows that, r = (P mi =1 σ i − i if r < s P m − i =1 σ i − i + 2 − m if r > s .where m := | A | and | A | = ( max { ≤ i ≤ k − , σ i = 1 } , if r < s max { ≤ i ≤ k − σ i = 0 } , if r > s ,with max( ∅ ) := 0.d) Indeed, write r as r = k X i =1 σ i − i , with σ i ∈ { , } k , and σ k = 1 . Put i − = max { ≤ i ≤ k − σ i = 1 } , and i + = max { ≤ i ≤ k − σ i = 0 } , with max( ∅ ) := 0. Then letting B − = B | i − and B + = B | i + , wededuce that (cid:8) ( B − , r − − k ) , ( B + , r + 2 − k ) (cid:9) ∈ E k − , and (cid:8) ( B, r ) , ( B − , r − − k ) (cid:9) , (cid:8) ( B, r ) , ( B + , r + 2 − k ) (cid:9) ∈ E k . The uniqueness is then clear since according to equation (2) it isnecessary that A ≺ B for { ( A, s ) , ( B, r ) } to be in E k , while equation(3) allows only two possible cardinalities for A . (cid:3) The graph G k introduced above is nothing else but a concrete representa-tion of the countably branching diamond graph of depth k defined recursivelyin the previous section. Proposition 2.1.
For all k ≥ , G k is graph isomorphic to D ωk .Proof. Recall that D ωk := ( V ωk , E ωk ) and G k := ( V k , E k ). We will think of s ( D ωk ) (resp. t ( K ,ω )) to be the bottom vertex (resp. the top vertex) of D ωk ,we shall use the notation b ωk for s ( D ωk ) and t ωk for t ( D ωk ). We shall prove byinduction on k ∈ N the following statement: H k : there exists a graph isomorphism ϕ k : D ωk → G k such thati) for all n ≤ k − ϕ k | D ωn = ϕ n ,ii) and if e ω = { u, v } ∈ E ωk , and if ( A, r ) = ϕ k ( u ) and ( B, s ) = ϕ k ( v ) then r = s + 2 − k .For the base case, define the map ϕ : V ω → V , t ω ( ∅ , b ω ( ∅ , ϕ statisfies H .Now assume that ( H k ) holds and recall that V ωk +1 = V ωk ∪ (cid:8) ( e ω , j ) : e ω ∈ E ωk , j ∈ N (cid:9) . We define, as required for the old vertices, ϕ k +1 ( v ) = ϕ k ( v ) if v ∈ V ωk . For the new vertices ( e ω , j ) where e ω = { u, v } ∈ E ωk and j ∈ N wechoose ϕ k +1 (( e ω , j )) as follows. If ϕ k ( u ) = ( A, r ) and ϕ k ( v ) = ( B, s ), thenby the induction hypothesis (cid:8) ( A, r ) , ( B, s ) (cid:9) ∈ E k , and we can, using Lemma2.1 (c), assume without loss of generality that B = { b , b , . . . , b k } ∈ [ N ] k ,with b < b < . . . < b k , s ∈ B k , A ≺ B and r = s ± − k ∈ B m , with m < k .So we put ϕ k +1 (( e ω , j )) := (cid:16) B ∪ { b k + j } , r + s (cid:17) . First we note that since s ∈ B k and r ∈ B m , for some m < k , it followsthat r + s ∈ B k +1 . Thus ϕ k +1 (( e ω , j )) ∈ V k +1 \ V k . This shows that ϕ k +1 iswell defined, and that ϕ k +1 ( V ωk +1 \ V ωk ) ⊂ V k +1 \ V k . The claim that ϕ k +1 isbijective and that for u, v ∈ V ωk , { u, v } ∈ E ωk ⇐⇒ { ϕ k +1 ( u ) , ϕ k +1 ( v ) } ∈ E n ,can be now obtained from Lemma 2.1 (d) above, the definition of D ωk +1 andthe induction hypothesis. (cid:3) N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 11
Basic metric properties of the countably branching diamondgraphs.
In this section we discuss the basic metric properties of the count-ably branching diamond graphs which will be needed in the sequel.First of all, let d k denote the shortest path metric on D ωk . Since for twovertices ( A, r ) and (
B, s ) in V k to form an edge in D ωk it is necessary that | r − s | = 2 − k , it follows that(4) d k (cid:0) ( A, r ) , ( B, s ) (cid:1) ≥ | r − s | k . Secondly we observe that for any (
A, r ) ∈ V k (5) d k (cid:0) b ωk , ( A, r ) (cid:1) = r k and d k (cid:0) ( A, r ) , t ωk (cid:1) = (1 − r )2 k . Equality (5) can be generalized as follows. We say that two vertices (
A, r )and (
B, s ) in D ωk lie on the same vertical path if there exists a simpleincreasing path P = (cid:0) ( P , p ) , ( P , p ) , . . . , ( P n , p n ) (cid:1) of length n for some n ≤ k in D ωk , i.e., (cid:8) ( P m − , p m − ) , ( P m , p m ) (cid:9) ∈ E k for m = 1 , , . . . , n and p < p < . . . < p n (and thus p m = m − k + p ), such that ( A, r ) = ( P , p )and ( B, s ) = ( P n , p n ) or ( B, s ) = ( P , p ) and ( A, r ) = ( P n , p n ). In that casewe observe that P is the shortest path connecting ( P , p ) and ( P n , p n ) andthus(6) d k (cid:0) ( A, r ) , ( B, s ) (cid:1) = d k (cid:0) ( P n , p n ) , ( P , p ) (cid:1) = 2 k ( p n − p ) = 2 k | r − s | . Note that for k ∈ N we can write V k as V k = S ∞ j =1 V ( j, +) k ∪ S ∞ j =1 V ( j, − ) k where for j ∈ N , V ( j, +) k = n(cid:16) { j } ∪ A, r +12 (cid:17) : ( A, r ) ∈ V k − , < r < , and j < min( A ) o[ (cid:8) t ωk , ( { j } , ) (cid:9) V ( j, − ) k = n(cid:16) { j } ∪ A, r (cid:17) : ( A, r ) ∈ V k − , < r < , and j < min( A ) o[ (cid:8) b ωk , ( { j } , ) } . Let k ∈ N and i = j ∈ N . Note that V ( i, +) k ∩ V ( j, +) k = { t ωk } , V ( i, − ) k ∩ V ( j, − ) k = { b ωk } , and V ( j, +) k ∩ V ( j, − ) k = { ( j, ) } , V ( i, +) k ∩ V ( j, − ) k = ∅ . Secondly, there is no edge between any element of V ( i, +) k \ { t ωk } and anyelement of V ( j, +) k \ { t ωk } , any element of V ( i, − ) k \ { b ωk } and any element of V ( j, − ) k \ { b ωk } , any element of V ( i, +) k and any element of V ( j, − ) k , any elementof V ( i, +) k \ { ( i, ) } and any element of V ( i, − ) k \ { ( i, ) } . It follows thereforefrom (6) that, • if i = j ∈ N then for all ( A, r ) ∈ V ( i, +) k and ( B, s ) ∈ V ( j, +) k , d k (cid:0) ( A, r ) , ( B, s ) (cid:1) = d k (cid:0) ( A, r ) , t ωk (cid:1) + d k (cid:0) t ωk , ( B, s ) (cid:1) = (2 − r − s )2 k (7) • if i = j ∈ N then for all ( A, r ) ∈ V ( i, − ) k and ( B, s ) ∈ V ( j, − ) k , d k (( A, r ) , ( B, s )) = d k (cid:0) ( A, r ) , b ωk (cid:1) + d k (cid:0) b ωk , ( B, s ) (cid:1) = ( r + s )2 k (8) • if i = j ∈ N then for all ( A, r ) ∈ V ( i, +) k and ( B, s ) ∈ V ( j, − ) k , d k (cid:0) ( A, r ) , ( B, s ) (cid:1) =(9) min n d k (cid:0) ( A, r ) ,t ωk (cid:1) + d k (cid:0) t ωk , ( B, s ) (cid:1) , d k (cid:0) ( A, r ) , b ωk (cid:1) + d k (cid:0) b ωk , ( B, s ) (cid:1)o • if j ∈ N then for all ( A, r ) ∈ V ( j, +) k and ( B, s ) ∈ V ( j, − ) k , d k (cid:0) ( A, r ) , ( B, s ) (cid:1) = d k (cid:0) ( A, r ) , ( { j } , ) (cid:1) + d k (cid:0) ( { j } , ) , ( B, s ) (cid:1) (10) = ( r − )2 k + ( − s )2 k = ( r − s )2 k We can therefore deduce from (7)-(10) that if i = j ∈ N then for( A, r ) ∈ V ( i, +) k and ( B, s ) ∈ V ( j, − ) k , d k (cid:0) ( A, r ) , ( B, s ) (cid:1) = 2 k · min (cid:0) − r + + − s, r + s (cid:1) (11) = 2 k · min (cid:0) − r − s, r + s (cid:1) = 2 k ( − r − s if r + s ≥ r + s if r + s ≤ j ∈ N , let A + j (resp. A − j ) be the set obtained by adding (resp.substracting) j to each element of A , with the convention that ∅ ± j = ∅ .Define also s j ( A ) := { j } ∪ ( A + j ) whenever A = ∅ . Note that if A is anelement in [ N ] ≤ k for some k , then s j ( A ) belongs to [ N ] ≤ k +1 . For j ∈ N ,if A = { j, a , . . . , a m } we also define s − j ( A ) := A \ { j } − j and note that s − j ◦ s j ( A ) = A . Using the two combinatorial shifts s j and s − j one candefine (essentially) two natural isometries based on the self-similarities ofthe diamond graphs. The first two isometries are I ( j, − ) k : V ( j, − ) k → V k − (12) ( A, r ) (cid:0) s − j ( A ) , r (cid:1) , if ( A, r ) = b ωk and b ωk b ωk − , and I ( j, +) k : V ( j, +) k → V k − (13) ( A, r ) (cid:0) s − j ( A ) , r − (cid:1) , if ( A, r ) = t ωk and t ωk t ωk − , which are the canonical isometries from a lower (resp. upper) diamond in D ωk of depth k − D ωk − .The last isometry, F ( j, +) k : V ( j, +) k → V k − (14) ( A, r ) (cid:0) s − j ( A ) , − r ) (cid:1) , if ( A, r ) = t ωk and t ωk b ωk − , is an isometry of an upper diamond in D ωk of depth k − D ωk − thatflips the vertices upside down.3. Embeddability of the countably branching diamond graphs
Embeddability into Banach spaces containing particular ℓ ∞ -trees. In this section our main embedding theorem is proven. The embed-ding is based on the existence of certain trees in the target space. For thesake of clarity the study of the existence in Banach spaces of this technicaldevice is postponed to Section 3.2.
N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 13
Good ℓ ∞ -trees of arbitrary height. A linear ordering ( A i ) i ∈ N of the set[ N ] ≤ n is called compatible on [ N ] ≤ n if for any i, j in N :if max( A i ) < max( A j ) then i < j. (15)In other words, one can only assign the number j ∈ N to an element A ∈ [ N ] ≤ n , if for all elements B ∈ [ N ] ≤ n , with max( B ) < max( A ), were alreadycounted. It is easy to see that such a linear ordering exists and that always A = ∅ , and A = { } . Definition 3.1.
We say that a Banach space X contains ( C, D )-good ℓ ∞ -trees of arbitrary height if there are constants C, D ≥ n ∈ N , and any compatible linear ordering ( A i ) i ∈ N of [ N ] ≤ n there are a vector-tree ( x A ) A ∈ [ N ] ≤ n in S X and a functional-tree ( x ∗ A ) A ∈ [ N ] ≤ n in S X ∗ satisfyingthe following properties:(16) for all A, B ∈ [ N ] ≤ n , with max( B ) > max( A ) ,x ∗ A ( x A ) = 1 and x ∗ A ( x B ) = 0 , (17) for all ( a i ) ni =0 ⊂ R , and all B = { b , b , . . . , b n } ∈ [ N ] n , C k ( a i ) ni =0 k ∞ ≤ k a x ∅ + a x { b } + · · · + a n x { b ,...,b n } k X ≤ C k ( a i ) ni =0 k ∞ , (18) for every i ≤ j, every ( a m ) jm =0 ⊂ R , one has (cid:13)(cid:13)(cid:13) i X m =1 a m x A m (cid:13)(cid:13)(cid:13) X ≤ D (cid:13)(cid:13)(cid:13) j X m =1 a m x A m (cid:13)(cid:13)(cid:13) X . If a Banach space X contains for every ε >
0, (1 + ε, ε )-good ℓ ∞ -treesof arbitrary height, we say that X contains good ℓ ∞ -trees of arbitrary heightalmost isometrically.Note that condition (17) means that every branch ( x A ) A (cid:22) B is C -equivalentto the ℓ n +1 ∞ -unit vector basis, while condition (18) says that the sequence( x A i ) i ∈ N (where ( A i ) i ∈ N is the above chosen compatible linear ordering of[ N ] ≤ n ) is basic, with a basis constant not exceeding D . Remark . If X has a bimonotone FDD (or more generally if X embedsinto a space with a bimonotone FDD), then the sequence ( x A i ) i ∈ N can chosento be block sequence of that FDD (or an arbitrary small perturbation of ablock sequence), which implies that ( x A i ) i ∈ N is also bimonotone (or has abimonotonicity constant which is arbitrarily close to 1). Example . If X = c , then the vector-tree ( x A ) A ∈ [ N ] ≤ n together with thefunctional-tree ( x ∗ A ) A ∈ [ N ] ≤ n where x A = e max( A ) and x ∗ A = e ∗ max( A ) form a(1 , ℓ ∞ -tree of height n .3.1.2. The embedding.
Theorem 3.1.
Assume Y contains good ℓ ∞ -trees of arbitrary height almostisometrically, then for every ε > and every k ∈ N there exists Ψ k : D ωk → Y ,such that if x and y belong to the same vertical path then d k ( x, y ) ≤ (cid:13)(cid:13) Ψ k ( x ) − Ψ k ( y ) (cid:13)(cid:13) ≤ (1 + ε ) d k ( x, y ) , and if x and y do not belong to the same vertical path then d k ( x, y ) C ( ε ) ≤ (cid:13)(cid:13) Ψ k ( x ) − Ψ k ( y ) (cid:13)(cid:13) ≤ (1 + ε ) d k ( x, y ) , where C ( ε ) = 6(1 + ε ) .Moreover, if Y contains good ℓ ∞ -trees of arbitrary height almost isomet-rically and if ( y A i ) i ∈ N is bimonotone, where ( y A ) A ∈ [ N ] ≤ n is the vector-tree,and ( A i ) i ∈ N is a compatible linear ordering, then Ψ k can be defined such that C ( ε ) = 3 .Proof. Definition of the coefficients: For k ∈ N , we define inductively afamily of coefficients (cid:8) c k ( i, r ) : 0 ≤ i ≤ k, r ∈ S i ≤ m ≤ k B m (cid:9) ⊂ { , , . . . k } , asfollows:for k = 1, let c (0 ,
1) = 2, c (0 ,
0) = 0, and c (0 , ) = c (1 , ) = 1,and for k ≥ c k +1 ( i, r ), 0 ≤ i ≤ k + 1 and r ∈ S k +1 m = i B m will be chosenas follows: c k +1 (0 , r ) = r k +1 for all r ∈ k +1 [ m =0 B m and for i = 1 , . . . , k + 1 and r ∈ S k +1 m = i B m , we put c k +1 ( i, r ) = ( c k ( i − , r ) if 0 < r ≤ c k ( i − , − r )) if ≤ r < c k +1 ( i, r ) is well defined for all i = 0 , , . . . , k + 1, and r ∈ S k +1 m = i B j , since if r = , both formulae lead to the same term, and whenever r ∈ B m , for some i ≤ m ≤ k + 1, it follows that 2 r ∈ B m − , in case that r ≤ , and 2 − r ∈ B m − in case that r ≥ .Assume that Y contains good ℓ ∞ -trees of arbitrary height almost isomet-rically. We shall prove by induction on k ∈ N the following claim. Thetheorem follows easily. Claim 3.1.
For any ε > there is η ( ε ) ∈ (0 , ε ] such that for every (1 + η ( ε ) , η ( ε )) -good ℓ k ∞ -tree whose tree or vectors (resp. tree of functionnals)is ( y σ ) σ ∈ [ N ] ≤ k (resp. ( y ∗ σ ) σ ∈ [ N ] ≤ k ), the map Ψ k : D ωk → Y, ( A, r ) X D (cid:22) A c k ( | D | , r ) y D , has the following properties: Ψ k ( b ωk ) = 0 and Ψ k ( t ωk ) = 2 k y ∅ (19) y ∗∅ (cid:0) Ψ k ( A, r ) (cid:1) = r k for all ( A, r ) ∈ V ωk , (20) Ψ k is (1 + ε ) − Lipschitz , (21) if x, y ∈ V ωk belong to the same vertical path (cid:13)(cid:13) Ψ k ( x ) − Ψ k ( y ) (cid:13)(cid:13) Y ≥ d k ( x, y )(22) N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 15 if x, y ∈ V ωk do not belong to the same vertical path d k ( x, y ) C ( ε ) ≤ (cid:13)(cid:13) Ψ k ( x ) − Ψ k ( y ) (cid:13)(cid:13) Y , (23) where C ( ε ) = 6(1 + ε ) in full generality, and C ( ε ) = 3 if ( y A i ) i ∈ N is bimono-tone. We now proceed with the induction. For k = 1 we proceed as follows.Consider an (1 + ε, ε )-good ℓ ∞ -tree of height 1 in Y . Thus Ψ : D ω → Y ,is the following mapΨ ( b ω ) = 0 , Ψ ( t ω ) = 2 y ∅ , and Ψ ( { n } , ) = y ∅ + y { n } , for n ∈ N . (19) is clearly satisfied and (20) follows from condition (16).For m < n in N it follows from (16) that (cid:13)(cid:13) Ψ ( { n } , ) − Ψ ( { m } , ) (cid:13)(cid:13) Y = k y { n } − y { m } k Y ( ≤ d (cid:0) { m } , ) , ( { n } , ) (cid:1) , ≥ | y ∗{ m } ( y { n } − y { m } ) | = 1 . From (17) we deduce that1 ≤ (cid:13)(cid:13) Ψ ( t ω ) − Ψ ( { n } , ) (cid:13)(cid:13) Y = k y ∅ − y { n } k Y ≤ ε. Similiarly we verify that 1 ≤ (cid:13)(cid:13) Ψ ( b ω ) − Ψ ( { n } , ) (cid:13)(cid:13) Y ≤ ε . Since, more-over, k Ψ ( t ω ) − Ψ ( b ω ) k Y = 2 k y ∅ k = 2, we deduce that (21), (22) and (23)hold.Assume now that Claim 3.1 is true for some k ∈ N . Let ε > η ≤ η ( ε ), to be chosen small enough later, then for all pair of trees( y A ) A ∈ [ N ] ≤ k ⊂ S Y , ( y A ) A ∈ [ N ] ≤ k ⊂ S Y ∗ that form a (1 + η, η )-good ℓ k +1 ∞ -tree one has that (19)-(23) hold for ε .Fix now a (1 + η, η )-good ℓ k +1 ∞ -tree given by ( y A ) A ∈ [ N ] ≤ k +1 ⊂ S Y and ( y ∗ A ) A ∈ [ N ] ≤ k +1 ⊂ S Y ∗ , for some compatible linear ordering ( A n ) n ∈ N of[ N ] ≤ k +1 . For A = A m and B = A n , we write A < lin B if m < n .For each j ∈ N consider the trees ( y s j ( A ) ) A ∈ [ N ] ≤ k and ( y ∗ s j ( A ) ) A ∈ [ N ] ≤ k in S Y and S Y ∗ . In particular y s j ( ∅ ) = y { j } and y ∗ s j ( ∅ ) = y ∗{ j } .Define a binary relation < ( j )lin on [ N ] ≤ k by A < ( j )lin B if and only if s j ( A ) < lin s j ( B ). The following claim is straightforward. Claim 3.2.
For every j ∈ N , the binary relation < ( j ) lin is a compatible linearorder on [ N ] ≤ k for which the trees ( y s j ( A ) ) A ∈ [ N ] ≤ k and ( y ∗ s j ( A ) ) A ∈ [ N ] ≤ k , in S Y and S Y ∗ respectively, form an (1 + η, η ) -good ℓ k ∞ -tree. Therefore it follows from our induction hypothesis that for j ∈ N the mapΨ ( j ) k : V k → Y, ( A, r ) X D (cid:22) A c k ( | D | , r ) y s j ( D ) , satisfies conditions (19)-(23) with ( y s j ( A ) ) A ∈ [ N ] ≤ k and ( y s j ( A ) ) A ∈ [ N ] ≤ k insteadof ( y A ) A ∈ [ N ] ≤ k and ( y ∗ A ) A ∈ [ N ] ≤ k , and ε instead of ε . The map Ψ k +1 can actu-ally be written in terms of the maps Ψ ( j ) k . Indeed, if A = { j, a , . . . , a m } ∈ [ N ] ≤ k +1 \ {∅} and r ∈ B m , then Ψ k +1 ( A, r ) = X D (cid:22) A c k +1 ( | D | , r ) y D = r k +1 y ∅ + (P { j }(cid:22) D (cid:22) A c k ( | D | − , r ) y D if 0 < r ≤ P { j }(cid:22) D (cid:22) A c k ( | D | − , − r )) y D if ≤ r < r k +1 y ∅ + ( Ψ ( j ) k (cid:0) s − j ( A ) , r (cid:1) if 0 < r ≤ Ψ ( j ) k (cid:0) s − j ( A ) , − r ) (cid:1) if < r < . = r k +1 y ∅ + ( Ψ ( j ) k (cid:0) I ( j, − ) k (( A, r ))) if 0 < r ≤ Ψ ( j ) k (cid:0) F ( j, +) k (( A, r )) (cid:1) if < r < . (25) Verification of (19) and (20) : Elementary computations show thatΨ k +1 ( b ωk +1 ) = 0, whereas Ψ k +1 ( t ωk +1 ) = 2 k +1 y ∅ follows from (24). Verification of (21) : In order to verify the inequality (21) we can assumethat { ( A, r ) , ( B, s ) } ∈ E k +1 , and thus, there is a j ∈ N so that ( A, r ) , ( B, s ) ∈ V ( j, +) k +1 or ( A, r ) , ( B, s ) ∈ V ( j, − ) k +1 . Secondly, by Lemma 2.1 (b) it follows that | r − s | = 2 − ( k +1) , and | B | = k + 1 and A ≺ B , or | A | = k + 1 and B ≺ A . Letus first assume that ( A, r ) , ( B, s ) ∈ V ( j, +) k +1 , and, without loss of generalitythat A ≺ B , and | B | = k + 1. • If A = ∅ , and thus r = 1, ( A, r ) = t ωk +1 and 1 − s = 2 − ( k +1) , it followsfrom (24) that (cid:13)(cid:13) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) (cid:13)(cid:13) Y = (cid:13)(cid:13) (1 − s )2 k +1 y ∅ − Ψ ( j ) k ( F ( j, +) k ( B, s )) (cid:13)(cid:13) Y (26) ≤ (1+ η ) max (cid:8) , (cid:13)(cid:13) Ψ ( j ) k ( F ( j, +) k ( B, s ) (cid:1) k Y (cid:9) , where to get (26) we used first the upper bound in inequality (17),then the obvious fact that sup i ( | a i | ) = max {| a j | , sup i = j | a i |} , andfinally the lower bound in inequality (17). We are now in position touse the induction hypothesis (cid:13)(cid:13) Ψ ( j ) k ( F ( j, +) k ( B, s )) (cid:13)(cid:13) Y = (cid:13)(cid:13) Ψ ( j ) k ( F ( j, +) k ( B, s )) − Ψ ( j ) k ( b ωk ) (cid:13)(cid:13) Y = (cid:13)(cid:13) Ψ ( j ) k ( F ( j, +) k ( B, s )) − Ψ ( j ) k ( F ( j, +) k ( t ωk +1 ) (cid:13)(cid:13) Y ≤ (1 + ε d k (cid:0) F ( j, +) k ( B, s ) , F ( j, +) k ( t ωk +1 ) (cid:1) = (1 + ε d k +1 (cid:0) ( B, s ) , t ωk +1 (cid:1) = (1 + ε . • If A = ∅ , and thus j = min A , we deduce similarly from (24) and(17) that (cid:13)(cid:13) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s )) (cid:13)(cid:13) Y = (cid:13)(cid:13) ( s − r )2 − ( k +1) y ∅ + Ψ ( j ) k ( F ( j, +) k (( A, r ))) − Ψ ( j ) k ( F ( j, +) k (( B, s )) (cid:1) k Y ≤ (1+ η ) max { , k Ψ ( j ) k ( F ( j, +) k (( A, r ))) − Ψ ( j ) k ( F ( j, +) k (( B, s )) (cid:1) k Y } N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 17 and, using again the induction hypothesis (cid:13)(cid:13) Ψ ( j ) k ( F ( j, +) k (( A, r ))) − Ψ ( j ) k ( F ( j, +) k (( B, s )) (cid:1) k Y ≤ (1+ ε d k (cid:0) F ( j, +) k ( A, r ) , F ( j, +) k (( B, s )) (cid:1) ≤ (1+ ε d k +1 (cid:0) ( A, r ) , ( B, s ) (cid:1) = 1+ ε . Thus in both cases we obtain (cid:13)(cid:13) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s )) (cid:13)(cid:13) ≤ (1 + η ) (1 + ε ≤ ε. The case where (
A, r ) , ( B, s ) ∈ V ( j, − ) k +1 , for some j ∈ N , is obtained in asimilar way using the isometry I ( j, − ) k . Verification of (22) : We observe that whenever the vertices (
A, r ) and(
B, s ) belong to the same vertical path, one has d k +1 (cid:0) ( A, r ) , ( B, s ) (cid:1) =2 k +1 | r − s | but by (20) and (24),2 k +1 | r − s | = | y ∗∅ (cid:0) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) (cid:1) | ≤ (cid:13)(cid:13) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) (cid:13)(cid:13) Y , which proves (22). Verification of (23) : We introduce the following renorming of the sub-space Z = [ y A j : j ∈ N ] := span( y A j : j ∈ N ) of Y . Recall that by assump-tion ( y A n ) n ∈ N is a basis for Z whose basis constant is not greater than 1 + η .So for z = P ∞ n =1 a n y A n we put ||| z ||| = max m ≤ n (cid:13)(cid:13)(cid:13) n X j = m a j y A j (cid:13)(cid:13)(cid:13) Y . It is well known that ( y A i ) i ∈ N becomes bimonotone with respect to ||| · ||| andthat (cid:13)(cid:13)(cid:13) ∞ X n =1 a n y A n (cid:13)(cid:13)(cid:13) Y ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 a n y A n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( ⋆ ) ≤ η ) (cid:13)(cid:13)(cid:13) ∞ X n =1 a n y A n (cid:13)(cid:13)(cid:13) Y . We will now show inequality (23) for ||| · ||| with C ( ε ) = 3, and (23) for k · k Y will follow from ( ⋆ ). Let ˜ Y be the Banach space Z with the norm ||| · ||| , then( y A j ) n ∈ N is a bimonotone basis of ˜ Y . Claim 3.3.
The trees ( y A ) A ∈ [ N ] ≤ k +1 and ( z ∗ A ) A ∈ [ N ] ≤ k +1 , where z ∗ A = y ∗ A | Z for A ∈ [ N ] ≤ k are in S ˜ Y and S ˜ Y ∗ , respectively, and form an (1 + η, η ) -good ℓ k ∞ -tree for ˜ Y . The coordinate functionals, which we denote by (˜ y ∗ A j ) j ∈ N are also a basicbimonotone sequence in S ˜ Y ∗ . Remark . It is not necessarily true that ˜ y ∗ A = z ∗ A for all A ∈ [ N ] ≤ k +1 , butnevertheless the tree (˜ y ∗ A j ) j ∈ N is a tree in S ˜ Y ∗ , and form with ( y A j ) j ∈ N an(1 + η, η )-good ℓ k ∞ -tree for ˜ Y . First of all, for any (
A, r ) ∈ V k +1 it follows from (16), that ||| Ψ k +1 ( t ωk +1 ) − Ψ k +1 ( A, r ) ||| ≥ | ˜ y ∗∅ (Ψ k +1 ( t ωk +1 ) − Ψ k +1 ( A, r )) | = 2 k +1 (1 − r ) = d k +1 (cid:0) t ωk +1 , ( A, r ) (cid:1) , and ||| Ψ k +1 ( A, r ) − Ψ k +1 ( b ωk +1 ) ||| ≥ | ˜ y ∗∅ (Ψ k +1 ( A, r ) − Ψ k +1 ( b ωk +1 )) | = 2 k +1 r = d k +1 (cid:0) ( A, r ) , b ωk +1 (cid:1) . Thus, the following five cases remain to be taken care off: • If (
A, r ) , ( B, s ) ∈ V ( j, +) k +1 \ { t ωk +1 } , for some j ∈ N ,Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) =( r − s )2 k +1 y ∅ + Ψ ( j ) k ( F ( j, +) k ( A, r )) − Ψ ( j ) k ( F ( j, +) k ( B, s ) (cid:1) . And thus, using the bimonotonicity and the induction hypothesis,we deduce that ||| Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) ||| ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ ( j ) k ( F ( j, +) k (( A, r ))) − Ψ ( j ) k ( F ( j, +) k (( B, s )) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ d k (cid:0) F ( j, +) k (( A, r )) , F ( j, +) k (( B, s )) (cid:1) = 13 d k +1 (cid:0) ( A, r ) (cid:1) , ( B, s ) (cid:1) . • With similar arguments we can show that for j ∈ N and ( A, r ) , ( B, s ) ∈ V ( j, − ) k +1 \ { b ωk +1 } we have ||| Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) ||| ≥ d k +1 (cid:0) ( A, r ) (cid:1) , ( B, s ) (cid:1) . • If (
A, r ) ∈ V ( i, +) k +1 \ { t ωk +1 } and ( B, s ) ∈ V ( j, +) k +1 \ { t ωk +1 } , with i = j .Let us first note that since we have already shown (22) and since ||| · ||| ≥ k · k it follows that (22) is also satisfied for the norm ||| · ||| instead of k · k . Let us assume without loss of generality that s ≥ r .Since ˜ y ∗{ i } is the coordinate functional for y { i } (in Y ) it follows fromthe expression of Ψ k +1 in (24), and the definition of the coefficientsthat ||| Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) ||| ≥ | ˜ y ∗{ i } (cid:0) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s )) | = 2 k · − r ) = 2 k +1 (1 − r )= d k +1 (( A, r ) , t ωk +1 ) ≥ (cid:0) d k +1 (( A, r ) , t ωk +1 (cid:1) + d k +1 (cid:0) ( B, s ) , t ωk +1 ) (cid:1)(cid:1) = 12 d k +1 (cid:0) ( A, r ) , ( B, s ) (cid:1) . • If (
A, r ) ∈ V ( i, − ) k +1 \ { b ωk +1 } and ( B, s ) ∈ V ( j, − ) k +1 \ { b ωk +1 } , with i = j .This case can be treated with a similar argument, and the sameinequality is obtained. N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 19 • ( A, r ) ∈ V ( i, +) k +1 \ { t ωk +1 } and ( B, s ) ∈ V ( j, − ) k +1 \ { b ωk +1 } , with i = j .We apply the functionals ˜ y ∗{ i } , ˜ y ∗{ j } and ˜ y ∗∅ to the vector Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) and obtain from (24), (cid:13)(cid:13) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) (cid:13)(cid:13) ≥ | ˜ y ∗{ i } (cid:0) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) (cid:1) | = 2 k · − r ) = 2 k +1 (1 − r ) , (cid:13)(cid:13) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) (cid:13)(cid:13) ≥ | ˜ y ∗{ j } (cid:0) Ψ k +1 ( B, s ) − Ψ k +1 ( A, r ) (cid:1) | = 2 k s = 2 k +1 s, (cid:13)(cid:13) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) (cid:13)(cid:13) ≥ | ˜ y ∗∅ (cid:0) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) (cid:1) | = 2 k +1 ( r − s ) . Note that if r − s ≤ , and s ≤ , then 1 − r ≥ , and thus (cid:13)(cid:13) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) (cid:13)(cid:13) ≥
13 2 k +1 = 13 diam( V k +1 , d k +1 )which implies that (cid:13)(cid:13) Ψ k +1 ( A, r ) − Ψ k +1 ( B, s ) (cid:13)(cid:13) ≥ d k +1 (cid:0) ( A, r ) , ( B, s ) (cid:1) , and finishes the verification of the inequality (23) . (cid:3) Banach spaces containing good ℓ ∞ -trees of arbitrary height. The main goal of this section is give sufficient conditions for a Banach spaceto contain good ℓ ∞ -trees of arbitrary height almost isometrically. More pre-cisely we will show that if X is a separable reflexive Banach space whichhas an unconditional asymptotic structure, and with Sz( X ∗ ) > ω , then X contains good ℓ ∞ -trees of arbitrary height almost isometrically. The dif-ficulty is to get (1 + ε, ε )-good ℓ ∞ -trees of arbitrary height for every ε > C, D )-good ℓ ∞ -trees of arbitrary height for someconstant C, D ≥
1. A somehow technical and lengthy argument is thereforeneeded. It can be split into essentially three steps. First we will show thatif X is a separable reflexive Banach space with an unconditional asymptoticstructure, and if Sz( X ∗ ) > ω , then for all n ∈ N , ℓ n ∞ is in the asymptoticstructure of X up to some constant C ≥
1. Then we argue, using an asymp-totic analogue of a classical argument of James about the non-distortabilityof ℓ ∞ , that for all n ∈ N , ℓ n ∞ is in the asymptotic structure of X (i.e. withconstant C arbitraily close to 1). Finally, we prove that if for all n ∈ N ℓ n ∞ is in the asymptotic structure of X then X contains good ℓ ∞ -trees ofarbitrary height almost isometrically. In Section 3.2.1 we introduce all theingredients needed to carry out the proof of the main theorem which formsSection 3.2.2.3.2.1. Asymptotic properties and trees.
Unconditional asymptotic structure.
We recall the following notion which was introduced in [27]. Let X be aBanach space, and let us denote the cofinite dimensional subspaces of X by cof( X ). For n ∈ N let E be an n -dimensional Banach space with a normalized basis ( e j ) nj =1 . We say that E is in the n th asymptotic structureof X , or X contains E asymptotically and write E ∈ { X } n , if ∀ ε > ∀ X ∈ cof( X ) , ∃ x ∈ S X , . . . , ∀ X n ∈ cof( X ) , ∃ x n ∈ S X n such that(27) ( x j ) nj =1 is (1 + ε )-equivalent to ( e j ) nj =1 .For 1 ≤ p ≤ ∞ , we say that ℓ np is in the n -th asymptotic structure of X ,up to a constant C ≥
1, if for each n ∈ N there is an n -dimensional space E n , with a normalized basis ( e ( n ) j ) nj =1 , which is C -equivalent to the ℓ np -unitvector basis, so that E n ∈ { X } n , for all n ∈ N .We will need the following lemma which can be extracted from [27, 1.8.3] Lemma 3.1. If ℓ n ∞ is contained in the n -asymptotic structure of X up to aconstant C , then ℓ n ∞ is in the n -th asymptotic structure of X .Sketch of proof. It was observed that for any E ∈ { X } n , with a normalizedbasis ( e j ) nj =1 , and any subspace F of E which is spanned by a normalizedblock basis ( f j ) mj =1 of ( e j ) nj =1 is in { X } m . Secondly James’s result on thenon distordability of c implies that for any m ∈ N , C > ε there isan n := n ( m, C, ε ) so that any n -dimensional space E with a normed basis( e j ) nj =1 , which is C -equivalent to the ℓ n ∞ unit vector basis admits a blockbasis of length m which is (1 + ε )-equivalent to the ℓ m ∞ unit vector basis.The conclusion follows. (cid:3) Remark . Actually, although not needed in this paper, the same is truefor all 1 ≤ p ≤ ∞ , and follows from the quantitative version of Krivine’sTheorem proven by Rosenthal in [43, Theorem 3.6]The following property will be crucially used in the sequel. Definition 3.2.
We say that a Banach space has an unconditional asymp-totic structure with constant C ≥
1, or a C -unconditional asymptotic struc-ture, if for all n ∈ N ∃ X ∈ cof( X ) , ∀ x ∈ S X , . . . , ∃ X n ∈ cof( X ) , ∀ x n ∈ S X n so that(28) ( x j ) nj =1 is C -unconditional. Remark . Having and unconditional asymptotic structure for a Banachspace is strictly weaker than having an unconditional basic sequence. Forinstance, the Argyros-Delyianni space [1] does not have any unconditionalbasic sequence but has an unconditional asymptotic structure.As noted in [27] the property of having an unconditional asymptotic struc-ture or containing a finite dimensional space E asymptotically can be de-scribed in the language of a game between two players. Assume that for n ∈ N the cofinite player picks a cofinite dimensional subspace X and thenthe vector player an element x ∈ S X . The players repeat this proce-dure n times to obtain cofinite dimensional subspaces X , X , . . . , X n andand vectors x , x , . . . , x n . The cofinite player has won if the sequence x , x , . . . , x n is a C -unconditional sequence. It follows therefore that X has an C -unconditional asymptotic structure if and only if for every n ∈ N the cofinite player has a winning strategy in that game. For n ∈ N an n -dimensional space E with a normalized basis ( e j ) nj =1 E is in the n -dimensional asymptotic structure of X if and only if for every ε > N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 21 vector player has a winning strategy for every ε >
0, if his or her goal is toobtain a sequence ( x j ) nj =1 which is (1 + ε )-equivalent to ( e j ) nj =1 . Tree reformulation for spaces with separable dual.
In this paper we will only consider trees of finite height which are countablybranching, i.e. families of the form ( x A ) A ∈ [ N ] ≤ n indexed by [ N ] ≤ n , for some n ∈ N .Let X be a Banach space and let for some n ∈ N , ( x A ) A ∈ [ N ] ≤ n be atree in X . The tree ( x A ) A ∈ [ N ] ≤ n is said to be normalized if ( x A ) A ∈ [ N ] ≤ n ⊂ S X . A node of ( x A ) A ∈ [ N ] ≤ n ⊂ S X is a sequence of the form (cid:8) x A ∪{ n } : n ∈ N , n > max( A ) (cid:9) , where A ∈ [ N ] A Banach space X is said to have the C -unconditional finitetree property, if for any n ∈ N , any normalized weakly null tree ( x A ) A ∈ [ N ] ≤ n in X has a branch which is C -unconditional.It is well-known that if a Banach space has a separable dual , a propertydefined in terms of a game can be reformulated in terms of containment ofcertain countably branching trees whose branches reflect the desired prop-erty. Lemma 3.2. If X ∗ is separable, then X has a C -unconditional asymptoticstructure if and only if X has the C -unconditional finite tree property.Proof. In the case that X ∗ is separable, and the sequence ( x ∗ n ) n ∈ N ⊂ S X isdense in S ∗ X , we can replace in (27) and (28) the set cof( X ) by the countableset of cofinite dimensional subspaces ( Y n ) n ∈ N , with Y n = N ( x ∗ , x ∗ , . . . x ∗ n ) = { x ∈ X : x ∗ j ( x ) = 0 for j = 1 , , . . . , n } . From this fact the conclusion followsfrom [32, Theorem 3.3] or [33, Corollary 1]. (cid:3) The next lemma can be proved along the same line and its proof is omit-ted. Lemma 3.3. If X ∗ is separable, then for any n ∈ N and any n -dimensionalspace E with normalized basis ( e j ) nj =1 , E is in the n -th asymptotic structureof X if and only if for every ε > there is a weakly null tree ( x A ) A ∈ [ N ] ≤ n in S X all of whose branches are (1 + ε ) -equivalent to ( e j ) nj =1 . Moreover, in thatcase ( x A ) A ∈ [ N ] ≤ n can be chosen inside every cofinite dimensional subspace. Sufficient conditions for the containment of good ℓ ∞ -trees. In thissection we give sufficient conditions that guarantee the presence of good ℓ ∞ -trees of arbitrary height. The good ℓ ∞ -trees of arbitrary height are obtainedby pruning carefully certain trees. The procedure of pruning a tree is whatcorresponds to the action of taking a subsequence for a sequence. Formallyspeaking we define a pruning of ( x A ) A ∈ [ N ] ≤ n as follows. Let π : [ N ] ≤ n → [ N ] ≤ n be an order isomorphism with the property that if F ∈ [ N ] We say that a Banach space X contains ( C, D )-good ℓ ∞ -trees of arbitrary height if there are constants C, D ≥ n ∈ N , and any compatible linear ordering ( A i ) i ∈ N of [ N ] ≤ n there are a vector-tree ( x A ) A ∈ [ N ] ≤ n in S X and a functional-tree ( x ∗ A ) A ∈ [ N ] ≤ n in S X ∗ satisfyingthe following properties:(16) for all A, B ∈ [ N ] ≤ n , with max( B ) > max( A ), x ∗ A ( x A ) = 1 and x ∗ A ( x B ) = 0 , (17) for all ( a i ) ni =0 ⊂ R , and all B ∈ [ N ] n ,1 C k ( a i ) ni =0 k ∞ ≤ k a x ∅ + a x { b } + · · · + a n x { b ,...,b n } k X ≤ C k ( a i ) ni =0 k ∞ , (18) for every i ≤ j , every ( a m ) jm =0 ⊂ R , one has k i X m =1 a m x A m k X ≤ D k j X m =1 a m x A m k X , If X ∗ is separable and if ℓ n ∞ is in the n -th asymptotic structure of X upto a constant D ≥ 1, then by Lemma 3.3 there is a normalized weakly treeall of whose branches are D (1 + ε )-equivalent to the canonical basis of ℓ n +1 ∞ .Obviously, such a tree (and any of its prunings) satisfies condition (17), but(16) and (18) might not hold. In the next lemma we show that for anycompatible linear ordering there exists a pruning such that (16) and (18)hold. N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 23 Lemma 3.4. Let X ∗ be separable. Assume that for all n ∈ N , ℓ n ∞ is inthe n -th asymptotic structure of X up to a constant D ≥ . Then for each ε > , X contains ( D (1 + ε ) , ε ) -good ℓ ∞ -trees of arbitrary height. Inparticular, if ℓ n ∞ is in the n -th asymptotic structure of X , then X containsgood ℓ ∞ -trees of arbitrary height almost isometrically.Proof. Let n ∈ N , ε > 0, and let ( A j ) j ∈ N be a compatible linear ordering of[ N ] ≤ n . Since ℓ n ∞ is in the n -th asymptotic structure of X up to a constant D ≥ X ∗ is separable, by Lemma 3.3 there is a weakly null tree( y A ) A ∈ [ N ] ≤ n in X such that k y A k = 1 all of whose branches are D (1 + ε )-equivalent to the canonical basis of ℓ n +1 ∞ . Moreover, ( y A ) A ∈ [ N ] ≤ n can bechosen inside every cofinite dimensional subspace. Note that (17) is satisfiedfor the tree ( y A ) A ∈ [ N ] ≤ n . We choose 0 < δ < ε small enough so that thefollowing condition holds:If ( v j ) n +1 j =1 is a normalized basic sequence, whose basis constant(29) is not larger than (1 + ε ), then any sequence ( u i ) n +1 i =1 , for which k u j − v j k ≤ δ , j = 1 , . . . , n , is √ ε -equivalent to ( v j ) n +1 j =1 . X ∗ being separable, pick ( f ∗ j ) j ∈ N a dense sequence in S X ∗ . Inductivelywe will choose for every j ∈ N , an element ˜ A j ∈ [ N ] ≤ n , a finite set F j ⊂ S X ∗ ,and an element ˜ y j ∈ B X so that˜ A = ∅ , and if A j = A i ∪ { max( A j ) } , for some i < j , then(30) ˜ A j = ˜ A i ∪ { m } , with m > max { ˜ A s : s < j } , { f ∗ , f ∗ , . . . , f ∗ j } ⊂ F j and there is an ˜ y ∗ j ∈ F j , with ˜ y ∗ j (˜ y j ) = k ˜ y j k ,(31) max f ∗ ∈ F j | f ∗ ( x ) | ≥ 11 + δ k x k for all x ∈ span(˜ y i ) ji =1 ,(32) f ∗ (˜ y j ) = 0 for all f ∗ ∈ F i , with i < j ,(33) k ˜ y j − y ˜ A j k < δ, and k ˜ y j k = 1 . (34)For j = 1, we put ˜ y = y ∅ , choose ˜ y ∗ ∈ S X ∗ so that ˜ y ∗ (˜ y ) = 1, andput F = { f ∗ , ˜ y ∗ } , (30)-(34) is then clearly satisfied. Now assume that forsome j ∈ N we have chosen ˜ A i , F i and ˜ y i , for i = 1 , , . . . , j . First we notethat (32) and (33) implies that the normalized sequence (˜ y i ) ji =1 is a basicsequence which constant not exceeding 1 + δ . Since the linear ordering iscompatible we can write A j as A j = A i ∪ { t } with i < j and t > max( A i )and let m = max s Szlenk index . Thedefinition of the Szlenk derivation and the Szlenk index have been firstintroduced in [44] and we refer to [24] or [33] for a thorough account onthis central notion in asymptotic Banach space theory. Consider a realseparable Banach space X and K a weak ∗ -compact subset of X ∗ . For ε > V be the set of all relatively weak ∗ -open subsets V of K such thatthe norm diameter of V is less than ε and s ε ( K ) = K \ ∪{ V : V ∈ V} . We define inductively s αε ( K ) for any ordinal α , by s α +1 ε ( K ) = s ε ( s αε ( K ))and s αε ( K ) = ∩ β<α s βε ( K ) if α is a limit ordinal. Then we define Sz( X, ε )to be the least ordinal α so that s αε ( B X ∗ ) = ∅ , if such an ordinal exists.Otherwise we write Sz( X, ε ) = ∞ . The Szlenk index of X is finally definedby Sz( X ) = sup ε> Sz( X, ε ) . We denote ω the first infinite ordinal and ω the first uncountable ordinal. The Szlenk index, in particular its utilizationin the next lemma, is the pivotal notion in this article since it allows usto establish a bridge between the embeddabilty results from Section 3 andthe non-embeddability results from Section 4. Indeed, it follows from atheorem of Knaust, Odell and Schlumprecht [23] that a separable Banachspace admits an equivalent asymptotically uniformly smooth norm if andonly if Sz( X ) ≤ ω . Then it is easy to see that for a reflexive Banach spacethe condition Sz( X ∗ ) ≤ ω is equivalent to the existence of an equivalentasymptotically uniformly convex norm on X . With this information athand, we can almost forget the formulations in terms of renormings andwork essentially with the notion of the Szlenk index of a Banach space.To prove Theorem 3.2 below it would be sufficient to show that a separablereflexive Banach space X , that has a C -unconditional asymptotic structurefor some C ≥ X ∗ ) > ω , contains ℓ n ∞ in its n -th asymptoticstructure up to some constant. Modifying slightly the proof of Lemma 3.4one can actually prove a little bit more. Lemma 3.5. Let X be a separable reflexive Banach space. Assume that X has a C -unconditional asymptotic structure for some C ≥ and that Sz( X ∗ ) > ω . Then, there is η > so that for all ε > , X contains ( √ ε Cη , ε ) -good ℓ ∞ -trees of arbitrary height. In particular, there is η > so that for all ε > and every n ∈ N , ℓ n ∞ is in the n -th asymptoticstructure of X up to constant √ ε Cη . N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 25 Proof. Without loss of generality (simply pass to an appropriate cofinitedimensional subspace X if needed) we can assume that ∀ x ∈ S X , ∃ X ∈ cof( X ) , , . . . , ∃ X n +1 ∈ cof( X ) , ∀ x n +1 ∈ S X n +1 so that(35)( x j ) n +1 j =1 is C -unconditional.Since X is separable and reflexive with Sz( X ∗ ) > ω , we can apply [30,Theorem 5.1, (i) ⇒ (iii)] to the space X ∗ , and conclude, that there is an η > 0, so that for each k ∈ N there is a tree ( x ( k ) A ) A ∈ [ N ] ≤ k ⊂ B X and aweakly null tree ( x ( k ) ∗ A ) A ∈ [ N ] ≤ k ⊂ B X ∗ with the following properties: • x ( k ) ∗ A ( x ( k ) B ) > η , for all A, B ∈ [ N ] ≤ k \ {∅} , with A (cid:22) B , • (cid:12)(cid:12) x ( k ) ∗ A ( x ( k ) B ) (cid:12)(cid:12) < η/ 2, for all A, B ∈ [ N ] ≤ k \ {∅} , with A B , • w-lim n →∞ x ( k ) A ∪{ n } = x ( k ) A , for all A ∈ [ N ] 0, and let ( A j ) j ∈ N be a compatible linear ordering of[ N ] ≤ n . Because we are now dealing with semi-normalized trees, we modifyslightly condition (29) as follows. We choose 0 < δ < ε small enough so thatthe following condition holds:(47 ′ ) If ( v j ) n +1 j =1 is a basic sequence with k v j k ∈ [ η/ , j = 1 , , . . . , n +1,whose basis constant is not larger than (1 + ε ), then any sequence ( u i ) n +1 i =1 ,for which k u j − v j k ≤ δ for every j = 1 , . . . , n + 1, is √ ε -equivalent to( v j ) n +1 j =1 .With a similar argument we can choose inductively, for every j ∈ N , anelement ˜ A j ∈ [ N ] ≤ n , a set F j , an element ˜ y j ∈ B X satisfying (30)-(33), and(52 ′ ) k ˜ y j − y ˜ A j k < δ k y ˜ A j k , and k ˜ y j k ∈ [ η/ , , and some crucial additional condition. Before stating this last condition wenote that from (30) it follows that I j = { s : A s (cid:22) A j } ⊂ { , , . . . , j } andwe put l := | I j | = | A j | . The last condition is: ∃ X l +1 ∈ cof( X ) , ∀ y l +1 ∈ S X l +1 , . . . , ∃ X k ∈ cof( X ) , ∀ y k ∈ S X k so that(37) { ˜ y s : s ∈ I j } ∪ { y l + i : i = 1 , . . . , k − l } is C -unconditional . Note that if l = k , then (37) means that { ˜ y s : s ∈ I j } is C -unconditional.The base case is straightforward. Now assume that for some j ∈ N wehave chosen ˜ A i , F i and ˜ y i , for i = 1 , , . . . , j . Using (37) we can find aspace ˜ X ∈ cof( X ) so that (with I j and l as defined before (37)) ∀ y l +1 ∈ S ˜ X , ∃ X l +2 ∈ cof( X ) , ∀ y l +2 ∈ S X l +2 , . . . , ∃ X k ∈ cof( X ) , ∀ y k ∈ S X k and { ˜ y s : s ∈ I j } ∪ { y l + i : i = 1 , . . . k − l } is C -unconditional. If l = n , we simply put˜ X = X . We define, ˜ A j +1 similarly using Z j = ˜ X ∩ n x ∈ X : f ∗ ( x ) = 0 for all f ∗ ∈ j [ i =1 F i o ∈ cof( X ) , instead of Y j , and thus (37) follow from the definition of Z j and the require-ment that y j +1 ∈ Z j . This finishes the inductive step.We define for A ∈ [ N ] ≤ n , z A = ˜ y j k ˜ y j k , and z ∗ A = ˜ y ∗ j , where j ∈ N is suchthat π ( A ) = ˜ A j . It remains to prove that (17) holds. We first note that forany B in [ N ] n +1 , say B = A j , it follows that ( z E ) E (cid:22) B = (cid:16) ˜ y i k ˜ y i k (cid:17) i ≤ j, A i (cid:22) A j is(1 + ε )-unconditional by (37). Letting I = { i ≤ j : A i (cid:22) A j } , we deducemax n(cid:13)(cid:13)(cid:13) X E (cid:22) B ξ E x E (cid:13)(cid:13)(cid:13) : ( ξ E ) E (cid:22) B ⊂ [ − , o ≤ η max n(cid:13)(cid:13)(cid:13) X i ∈ I ξ i ˜ y i (cid:13)(cid:13)(cid:13) : ( ξ i ) i ∈ I ∈ [ − , I o (since k ˜ y i k ≥ η/ i ∈ I )= 4 η max n(cid:13)(cid:13)(cid:13) X i ∈ I ξ i ˜ y i (cid:13)(cid:13)(cid:13) : ( ξ i ) i ∈ I ∈ {− , } I o (by convexity we onlyneed to consider the extreme points of [ − , I which are {− , } I )= 4 Cη (cid:13)(cid:13)(cid:13) X i ∈ I ˜ y i (cid:13)(cid:13)(cid:13) (by (37)) ≤ √ ε Cη (cid:13)(cid:13)(cid:13) X E (cid:22) π ( B ) y E (cid:13)(cid:13)(cid:13) (by (29) and (34))= √ ε Cη k x ( n +1) π ( B ) k (by (36)) ≤ √ ε Cη . Since ( z E ) E (cid:22) B is (1 + ε )-unconditional it also follows that (cid:13)(cid:13)(cid:13) X E (cid:22) B ξ E z E (cid:13)(cid:13)(cid:13) ≥ 11 + ε max E (cid:22) B | ξ E | , for all ( ξ E ) E (cid:22) B ⊂ [ − , (cid:3) Theorem 3.2. Let X be a separable reflexive Banach space. Assume that X has an unconditional asymptotic structure, and that Sz( X ∗ ) > ω , then X contains good ℓ ∞ -trees of arbitrary height almost isometrically.Proof. We will apply Lemma 3.5 and Lemma 3.4 successively as follows.First note that, by Lemma 3.5, for all n ∈ N ℓ n ∞ is in the n -th asymptoticstructure of X up to some constant D . By Lemma 3.1, for all n ∈ N , itfollows that ℓ n ∞ is in the n -th asymptotic structure of X . An appeal toLemma 3.4 finishes the proof. (cid:3) Embeddability into L . The fact that the countably branching dia-mond of depth 1 embeds isometrically into L is well known and can be foundimplicitly in Enflo’s (unpublished) argument showing that ℓ and L [0 , 1] arenot uniformly homeomorphic (cf. [6], [7], or [45]). The embedding is basedon the particular behavior of Rademacher functions in L [0 , similar embedding, for countably branching diamond graphs of arbi-trary depth, can be implemented without blowing up the distortion is not N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 27 completely obvious at first sight. However, using the results from [12] andan ultraproduct argument it can be shown that D ωk embeds into L withdistortion at most 2. In this section, an embedding using Bernoulli randomvariables (and no ultraproduct argument) that achieves the same distortionis given. Note that it follows from [26] that this distortion is optimal. It isworth noticing that in both our embedding proofs we start out with a tree( x A ) A ∈ [ N ] ≤ k , and in both cases the definition of of the image of ( A, r ) onlydepends on { x A | i : i = 1 , , . . . , | A |} . We start with the following technicallemma. Lemma 3.6. Let (Ω , Σ , P ) be an atomless probablity space. For every k ∈ N ,and every x ∈ V ωk , there exist measurable sets S k ( x ) ⊂ Ω such that (38) S k ( x ) ⊂ S k ( y ) whenever x, y ∈ V ωk lie on the same vertical path,and (39) S k ( x ) and S k ( y ) are independent if min( x ) = min( y ) . Proof. We are using the non recursive description of the diamond graph D ωk = ( V ωk , E ωk ). Let k ∈ N be fixed. We consider a (countable) family { ε A : A ∈ [ N ] ≤ k \{∅}} , indexed by the elements of [ N ] ≤ k \{∅} , of independentBernoulli random variables, i.e., P ( ε A = 1) = P ( ε A = − 1) = 1 / 2. For( A, r ) ∈ V ωk , we will define a measurable set S k ( A, r ) ⊂ Ω as follows. Weput S k ( ∅ , 0) = ∅ and S k ( ∅ , 1) = Ω. For ( A, r ) ∈ V ωk , with A = ∅ , say A = { a , a , . . . , a n } and r = P ni =1 σ i − i ∈ B n , ( i.e., σ i ∈ { , } , i = 1 , . . . , n − σ n = 1) we proceed as follows. For 1 ≤ i ≤ n , so that σ i = 1, define(40) T ik ( A, r ) := { ε A | i = 1 } ∩ \ m
A, r ) , ( B, s ) ∈ V ωk and { ( A, r ) , ( B, s ) } ∈ E ωk . As noticedpreviously it follows that r = s ± − k . We claim that S k ( A, r ) ⊂ S k ( B, s ), if r = s − − k , and that S k ( B, s ) ⊂ S k ( A, r ) if r = s +2 − k . In order to show ourclaim we can, as noted before, assume that B ∈ [ N ] k , s = P ki =1 σ i − i ∈ B k ,and that one of the two following cases holds: Case 1. r = s − − k and r = i − X i =1 σ i − i , and A = B | i − with i − = max { ≤ i ≤ k − σ i = 1 } if that maximum exists and otherwise A = ∅ and r = 0. Case 2. r = s + 2 − k and r = i + − X i =1 σ i − i + 2 − i + , and A = B | i + with i + = max { ≤ i ≤ k − σ i = 0 } if that maximum exists and otherwise A = ∅ and r = 1.In order to show our claim in the first case we can assume that i − exists,since otherwise S k ( A, r ) = ∅ . Since for every i ≤ i − , for which σ i = 1, wededuce that T ik ( A, r ) = { ε A | i = 1 } ∩ \ m
B, s ) lies above ( A, r ) on the same vertical path. From the inde-pendence condition of the family ( ε A ) A ∈ [ N ] ≤ k , and since for every ( A, r ) ∈ V ωk , with A = ∅ , the set S k ( A, r ) is in the σ -algebra generated by the fam-ily { ε C : C ∈ [ N ] ≤ k \ {∅} , min( C ) = min( A ) } it follows that S k ( A, r ) and S k ( B, s ) are independent if A, B ∈ [ N ] ≤ k \ {∅} , with min( A ) = min( B ). (cid:3) We finish our list of facts with a crucial observation which will allowus to prove our embedding result by induction. Assume that k ≥ j ∈ N , and put N j = { j + 1 , j + 2 , . . . } . For A ∈ [ N j ] k − and r = P ni =1 σ i − i ∈ B n , where n := | A | , (thus ( A, r ) ∈ V ωk − ), for 1 ≤ i ≤ n N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 29 we put T ( i ; j, +) k − ( A, r ) = T i +1 k ( { j } ∪ A, r ) and T ( i ; j, − ) k − ( A, r ) = T i +1 k ( { j } ∪ A, r ) . Note that r = + P ni =1 σ i − (1+ i ) and r = P ni =1 σ i − (1+ i ) , thus T ( i ; j, +) k − ( A, r )and T ( i ; j, − ) k − ( A, r ) are well defined, and we put S ( j, +) k − ( A, r ) := [ ≤ i ≤| A | ,σ i =1 T ( i ; j, +) k − ( A, r ) , and S ( j, − ) k − ( A, r ) := [ ≤ i ≤| A | ,σ i =1 T ( i ; j, − ) k − ( A, r ) . We define ˜ ε A = ε A | { ε { j } = − } for A ∈ [ N j ] k − . Then (˜ ε A ) A ∈ [ N j ] k − are in-dependent Bernoulli random variables on the probability space (˜ P j , ˜Σ j , ˜Ω j ),with ˜Ω j = { ε { j } = − } , ˜Σ j = Σ ∩ { ε { j } = − } , and for a measurable S ⊂ ˜Ω j , ˜ P j ( S ) = P ( S | { ε { j } = − } ) = 2 P ( S ). For A ∈ [ N j ] k − and r = P ni =1 σ i − i ∈ B n , where n = | A | , and i ≤ n , with σ i = 1, we note that T ( i ; j, +) k − ( A, r ) = T i +1 k ( { j } ∪ A, r )= { ε { j } = − } ∩ { ε { j }∪ A | i +1 = 1 } ∩ i \ m =2 ,σ m − =1 { ε { j }∪ A | m = − }∩ i \ m =2 ,σ m − =0 { ε { j }∪ A | m = 1 } = { ˜ ε A | i = 1 } ∩ \ m
For every k ∈ N there exists Ψ k : D ωk → L [0 , , such thatif x and y belong to the same vertical path then (44) (cid:13)(cid:13) Ψ k ( x ) − Ψ k ( y ) (cid:13)(cid:13) = d k ( x, y ) , and if x and y do not belong to the same vertical path then (45) d k ( x, y )2 ≤ (cid:13)(cid:13) Ψ k ( x ) − Ψ k ( y ) (cid:13)(cid:13) ≤ d k ( x, y ) . Proof. For k ∈ N and x ∈ V ωk let S k ( x ) ⊂ [0 , 1] be defined as in Lemma3.6, and define the map Ψ k : V ωk → L [0 , , x k χ S k ( x ) . For k = 0, our claim is trivially true. Assume that our claim holds for k ∈ N ∪ { } . The equality (44) follows immediately from (38) and (43),and the fact (6), shown in Section 2.3, that d k (cid:0) ( A, r ) , ( B, s ) (cid:1) = 2 k | s − r | , if ( A, r ) , ( B, s ), lie in the same vertical path. For general ( A, r ) and( B, s ) in V ωk +1 we proceed as follows. Note first that we can assume that( A, r ) , ( B, s ) ∈ V ωk +1 \ { b ωk +1 , t ωk +1 } , otherwise they would belong to the samevertical path. First let us assume that ( A, r ) and ( B, s ) are in V ( j, +) k +1 for some j ∈ N . Thus, write r and s as r = P Ai =1 σ i − i ∈ B A and s = P Bi =1 τ i − i ∈ B B . It follows that we can write A = { j } ∪ A ′ and B = { j } ∪ B ′ , with A ′ , B ′ ∈ [ N ] ≤ k , and r = r ′ +12 and s = s ′ +12 with r ′ = P A ′ i =1 σ ′ i − i ∈ B A ′ and s ′ = P A ′ i =1 τ ′ i − i ∈ B B ′ . Note that σ ′ i = σ i +1 , for i = 1 , , . . . , A ′ and τ ′ i = τ i +1 , for i = 1 , , . . . , B ′ . It follows therefore from the definitionof S k +1 ( A, r ) in (41) and the definition of the T ik +1 ( A, s ), i = 1 , , . . . , A ,with σ i = 1, in (40) that S k +1 ( A, r ) = (cid:8) ε { j } = 1 (cid:9) ∪ [ ≤ i ≤ A,σ i =1 T ik +1 ( A, r )= (cid:8) ε { j } = 1 (cid:9) ∪ [ ≤ i ≤ A ′ ,σ ′ i =1 T i +1 k +1 ( { j } ∪ A ′ , r ′ +12 )= (cid:8) ε { j } = 1 (cid:9) ∪ S ( j, +) k ( A ′ , r ′ )and, similarly S k +1 ( B, s ) = { ε { j } = 1 } ∪ S ( j, +) k ( B ′ , s ′ ) , where the sets S ( j, +) k ( A ′ , r ′ ), and S ( j, +) k ( B ′ , s ′ ) have been defined before thestatement of the theorem. It follows from the prior observations and theinduction hypothesis (using the probability space ( ˜Ω j , ˜Σ j , ˜ P j ) as definedabove) that2 k +1 (cid:13)(cid:13) χ S k +1 ( A,r ) − χ S k +1 ( B,s ) (cid:13)(cid:13) L [0 , = 2 k +1 (cid:13)(cid:13) χ S ( j, +) k ( A ′ ,r ′ ) − χ S ( j, +) k ( B ′ ,s ′ ) (cid:13)(cid:13) L [0 , = 2 k (cid:13)(cid:13) χ S ( j, +) k ( A ′ ,r ′ ) − χ S ( j, +) k ( B ′ ,s ′ ) (cid:13)(cid:13) L (˜ P j ) , but d k (cid:0) ( A ′ , r ′ ) , ( B ′ , s ′ ) (cid:1) ≤ k (cid:13)(cid:13) χ S ( j, +) k ( A ′ ,r ′ ) − χ S ( j, +) k ( B ′ ,s ′ ) (cid:13)(cid:13) L (˜ P j ) ≤ d k (cid:0) ( A ′ , r ′ ) , ( B ′ , s ′ ) (cid:1) . Using now (13) we conclude that d k +1 (cid:0) ( A, r ) , ( B, s ) (cid:1) ≤ (cid:13)(cid:13) Ψ k +1 (cid:0) ( A, r ) (cid:1) − Ψ k (cid:0) ( B, s ) (cid:1)(cid:13)(cid:13) L [0 , ≤ d k +1 (cid:0) ( A, r ) , ( B, s ) (cid:1) . If ( A, r ) and ( B, s ) are in V ( j, − ) k +1 for some j ∈ N , we deduce our claimsimilarly as in the previous case.If ( A, r ) ∈ V ( j, − ) k +1 and ( B, s ) ∈ V ( j, +) k +1 , for some j ∈ N (or vice versa), then( A, r ) and ( B, s ) lie on the same vertical path, a case we already handled. N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 31 If ( A, r ) ∈ V ( i, +) k +1 and ( B, s ) ∈ V ( j, +) k +1 , for i = j , we can assume that r ≥ s > and it follows from (2) and (39) that k χ S k +1 ( A,r ) − χ S k +1 ( B,s ) k L [0 , = k χ S ck +1 ( A,r ) − χ S ck +1 ( B,s ) k L [0 , = P (cid:0) S ck +1 ( A, r ) \ S ck +1 ( B, s ) (cid:1) + P (cid:0) S ck +1 ( B, s ) \ S ck +1 ( A, r ) (cid:1) = (1 − r ) + (1 − s ) − − r )(1 − s ) ≥ (1 − r ) + (1 − s ) − (1 − r )= 1 − s ≥ 12 2 − ( k +1) d k +1 (cid:0) ( A, r ) , ( B, s ) (cid:1) , and secondly (cid:13)(cid:13) χ S k +1 ( A,r ) − χ S k +1 ( B,s ) (cid:13)(cid:13) L [0 , = (cid:13)(cid:13) χ S ck +1 ( A,r ) − χ S ck +1 ( B,s ) (cid:13)(cid:13) L [0 , ≤ P ( S ck +1 ( A, r )) + P ( S ck +1 ( B, s ))= 2 − ( k +1) d k +1 (cid:0) ( A, r ) , ( B, s ) (cid:1) , which implies our claim in that case.If ( A, r ) ∈ V ( i, − ) k +1 and ( B, s ) ∈ V ( j, − ) k +1 , for i = j , we can proceed in asimilar way as in the previous case.If ( A, r ) ∈ V ( i, − ) k and ( B, s ) ∈ V ( j, +) k for i = j , we first assume that r + s ≤ (cid:13)(cid:13) χ S k +1 ( A,r ) − χ S k +1 ( B,s ) (cid:13)(cid:13) L [0 , = r + s − rs ≤ r + s = d k +1 (cid:0) ( A, r ) , ( B, s ) (cid:1) ( k +1) . Secondly the geometric-arithmetic mean inequality and the assumption that r + s ≤ r / s / ≤ r + s ≤ √ r + s , and thus 2 rs ≤ r + s which implies by (11) that r + s − rs ≥ r + s − ( k +1) d k +1 (cid:0) ( A, r ) , ( B, s ) (cid:1) , which implies our claim.If ( A, r ) ∈ V ( i, − ) k +1 and ( B, s ) ∈ V ( j, +) k +1 for i = j , and r + s ≥ 1, we canproceed similarly, which finishes our induction step and the proof of ourtheorem. (cid:3) Embeddability into L p ( Y ) . Recall that D k stands for the 2-branchingdiamond of depth k . In this section it is shown that one can build an embed-ding of D ωk into L p ( Y ) out of an embedding of D k into Y . This “embeddingtransfer” is useful in regards of some renorming problems that will be dis-cussed in Section 5. Theorem 3.4. If for every k ∈ N , D k embeds into Y with distortion at most C ≥ then for every k ∈ N , D ωk embeds into L p ([0 , , Y ) with distortionat most /p C . More precisely, for each k ≥ , there exists a -Lipschitz map ¯ φ k : D ωk → L p ([0 , , Y ) such that whenever x and y belong to the samevertical path in D ωk (46) k ¯ φ k ( x ) − ¯ φ k ( y ) k p ≥ C d k ( x, y ) , and whenever x and y do not belong to the same vertical path in D ωk (47) k ¯ φ k ( x ) − ¯ φ k ( y ) k p ≥ /p C . Proof. Let k · k p denotes the norm in L p ([0 , , Y ) and b k (resp. t k ) denotethe bottom vertex (resp. top vertex) of D k . Assume, as we may, that, foreach k ≥ 1, there exists φ k : D k → Y that satisfies for every x, y ∈ D k ,(48) 1 C d k ( x, y ) (37 a ) ≤ k φ k ( x ) − φ k ( y ) k Y (37 b ) ≤ d k ( x, y ) . We shall construct inductively, for each k ≥ 1, a 1-Lipschitz mapping¯ φ k : D ωk → L p ([0 , , Y ) satisfying the following conditions:for every x and y in D ωk belonging to the same vertical path(49) 1 C d k ( x, y ) ≤ k ¯ φ k ( x ) − ¯ φ k ( y ) k p , for every x and y in D ωk belonging to different vertical paths(50) 12 /p C d k ( x, y ) ≤ k ¯ φ k ( x ) − ¯ φ k ( y ) k p , for every x ∈ D ωk and u ∈ [0 , r ∈ D k satisfying(51) ¯ φ k ( x )( u ) = φ k ( r ) and d k ( b ωk , x ) = d k ( b k , r ) . One more time, we shall give ourselves once and for all a sequence ofindependent Bernoulli random variables defined on [0 , , 1] we will assume, as we may, that we will leave out countablyinfinitely many independent Bernoulli random variables from this sequence.The initialization case goes as follows. Note that D has four vertices: t (top), b (bottom), v l (left vertex), and v r (right vertex). Suppose we have amapping φ : D → Y satisfying (48). Note that D ω consists of a top vertex t ω , a bottom vertex b ω , and the sequence ( m j ) j ∈ N of midpoints of t ω and b ω . Let A := { ε j : j ∈ N } be a countably infinite collection of independentBernoulli random variables defined on [0 , φ : D ω → L p ([0 , , Y )as follows(52) ¯ φ ( b ω ) = φ ( b ) · χ [0 , , ¯ φ ( t ω ) = φ ( t ) · χ [0 , , and(53) ¯ φ ( m j ) = φ ( v l ) · χ { ε j =0 } + φ ( v r ) · χ { ε j =1 } . Condition (51) is clearly statisfied and using (37b) it is easily checkedthat the map is 1-Lipschitz. An appeal to (37 a ) will show that1 C d ( x, y ) ≤ k ¯ φ ( x ) − ¯ φ ( y ) k p , N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 33 holds whenever x and y belong to the same vertical path in D ω , which proves(49). If x = m i and y = m j , for some i < j , then elementary computationsshow that k ¯ φ ( m i ) − ¯ φ ( m j ) k pp ≥ P ( ε i = ε j ) k φ ( v l ) − φ ( v r ) k pp = 12 k φ ( v l ) − φ ( v r ) k pp ≥ C p d ( v l , v r ) p = 12 C p d ( m i , m j ) p , which gives (50).To carry out the inductive step, further notation is needed. Similarly tothe notation for the countably branching diamond graph, let D ( l, +) k , D ( r, +) k , D ( l, − ) k , and D ( r, − ) k denote the “top left”, “top right”, “bottom left”, and“bottom right” subdiamonds of D k +1 of depth k . We identify each of themwith D k via the canonical isometry which preserves “top”, “bottom”, “left”and “right” for all subdiamonds. Let φ k +1 satisfy (48) (where k is substi-tuted with k + 1) and let φ ( l, +) k , φ ( r, +) k , φ ( l, − ) k , and φ ( r, − ) k be the restrictionsof φ k +1 to D ( l, +) k +1 , D ( r, +) k +1 , D ( l, − ) k +1 , and D ( r, − ) k +1 respectively. Via the identifica-tion described above we regard φ ( l, +) k , φ ( r, +) k , φ ( l, − ) k , and φ ( r, − ) k as mappingsfrom D k into Y satisfying (48). By the induction hypothesis, there are maps¯ φ ( l, +) k , ¯ φ ( r, +) k , ¯ φ ( l, − ) k , and ¯ φ ( r, − ) k from D ωk into L p ([0 , , Y ) satisfying (49) and(50). Let ( m j ) j ∈ N be the midpoints of t ωk +1 and b ωk +1 in D ωk +1 and take asequence ( ε j ) j ∈ N of independent random Bernoulli variables on [0 , φ k +1 : D ωk +1 → L p ([0 , , Y ) as follows:¯ φ k +1 ( x ) := ( ¯ φ ( l, +) k ( x ) · χ { ε j =0 } + ¯ φ ( r, +) k ( x ) · χ { ε j =1 } , if x ∈ V ( j, +) k +1 for some j , ¯ φ ( l, − ) k ( x ) · χ { ε j =0 } + ¯ φ ( r, − ) k ( x ) · χ { ε j =1 } , if x ∈ V ( j, − ) k +1 for some j . Note that ¯ φ ( l, +) k ( x ) (resp. ¯ φ ( l, +) k ( x ), ¯ φ ( l, +) k ( x ), ¯ φ ( l, +) k ( x ), ¯ φ ( l, +) k ( x )) are well-defined when we identify, as we may, V ( j, +) k +1 , and V ( j, − ) k +1 with D ωk . Moreover,¯ φ k +1 is well-defined as the two formulas coincide for x = m j . Condition (51)is clearly satisfied. We now verify (49)-(50) and the 1-Lipschitz conditionfor ¯ φ k +1 . Case 1: If x, y ∈ V ( j, +) k or x, y ∈ V ( j, − ) k for some j ∈ N . We onlytreat the case x, y ∈ V ( j, +) k since the case x, y ∈ V ( j, − ) k is completelysimilar. k ¯ φ k +1 ( x ) − ¯ φ k +1 ( y ) k pp (54) = 12 ( k ¯ φ ( l, +) k ( x ) − ¯ φ ( l, +) k ( y ) k pp + k ¯ φ ( r, +) k ( x ) − ¯ φ ( r, +) k ( y ) k pp ) . It now follows easily from (54) and the induction hypothesis, that if x and y belong to the same vertical path then1 C d k ( x, y ) ≤ k ¯ φ k ( x ) − ¯ φ k ( y ) k p , and if x and y belong to different vertical paths12 /p C d k ( x, y ) ≤ k ¯ φ k ( x ) − ¯ φ k ( y ) k p . Moreover, if x and y form a pair of adjacent vertices in D ωk +1 theneither x, y ∈ V ( j, +) k or x, y ∈ V ( j, − ) k , and combining (37b) and (54)one gets k ¯ φ k +1 ( x ) − ¯ φ k +1 ( y ) k p ≤ d k +1 ( x, y ) = 1 , which is sufficient to show that ¯ φ k +1 is 1-Lipschitz since the domainspace is an unweighted graph equipped with the shortest path metric. Case 2: If x ∈ V ( j, +) k and y ∈ V ( j, − ) k . In this case x and y belong to thesame vertical path. First consider u ∈ { ε j = 0 } . By the inductionhypothesis one can assume that ¯ φ k +1 ( x )( u ) = φ k +1 ( r ) for some r ∈ D ( l, +) k +1 , with d k +1 ( b k +1 , r ) = d k +1 ( b ωk +1 , x ), and ¯ φ k +1 ( y )( u ) = φ k +1 ( s )for some s ∈ D ( l, − ) k +1 , with d k +1 ( b k +1 , s ) = d k +1 ( b ωk +1 , y ). Then, k ¯ φ k +1 ( x )( u ) − ¯ φ k +1 ( y )( u ) k Y = k φ k +1 ( r ) − φ k +1 ( s ) k Y ≥ C d k +1 ( r, s )= 1 C d k +1 ( x, y ) . A similar inequality is obtained if u ∈ { ε j = 1 } . To conclude itremains to observe that k ¯ φ k +1 ( x ) − ¯ φ k +1 ( y ) k p ≥ inf u ∈ [0 , k ¯ φ k +1 ( x )( u ) − ¯ φ k +1 ( y )( u ) k Y . Case 3: If x ∈ V ( i, ± ) k +1 and y ∈ V ( j, ± ) k +1 for some i = j ∈ N . In thiscase x and y do not belong to the same vertical path. There arefour possible choices of ± signs, but the argument is essentially thesame in each case. So let us consider the case of x ∈ V ( i, +) k +1 and y ∈ V ( j, +) k +1 for some i = j ∈ N . We will proceed as in Case 2 andlook at pointwise inequalities. There are four subcases to consider. If u ∈ { ε i = 0 }∩ { ε j = 1 } . By the induction hypothesis one can assumethat ¯ φ k +1 ( x )( u ) = φ k +1 ( r ) for some r ∈ D ( l, +) k +1 , with d k +1 ( b k +1 , r ) = d k +1 ( b ωk +1 , x ), and ¯ φ k +1 ( y )( u ) = φ k +1 ( s ) for some s ∈ D ( r, +) k +1 , with d k +1 ( b k +1 , s ) = d k +1 ( b ωk +1 , y ). Using (37a), we get k ¯ φ k +1 ( x )( u ) − ¯ φ k +1 ( y )( u ) k Y = k φ k +1 ( r ) − φ k +1 ( s ) k (55) ≥ C d k +1 ( r, s ) = 1 C d k +1 ( x, y ) . A similar argument shows that (55) also holds for u ∈ { ε i = 1 }∩{ ε j =0 } . Hence k ¯ φ k +1 ( x ) − ¯ φ k +1 ( y ) k pp ≥ P ( ε i = ε j ) d k +1 ( x, y ) p C p = d k +1 ( x, y ) p C p , which gives (24a). (cid:3) N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 35 Corollary 3.1. Suppose ≤ p < ∞ , and let Y be a non-superreflexiveBanach space. Then, for every ε > and k ∈ N , D ωk admits a bi-Lipschitzembedding into L p ([0 , , Y ) with distortion at most /p + ε .Proof. By a recent result of Pisier [39], which refines a result of Johnsonand Schechtman [15], for every ε > φ k : D k → Y such that, for all x, y ∈ D k ,12 + ε d k ( x, y ) ≤ k φ k ( x ) − φ k ( y ) k Y ≤ d k ( x, y ) . The conclusion follows from Theorem 3.4. (cid:3) Non-embeddability of the countably branching diamondgraphs In a celebrated unpublished article (cf. [6], [7], or [45]), Enflo used anapproximate midpoint argument to show that L and ℓ are not uniformlyhomeomorphic. This simple, but clever and extremely useful argument, isbased on the fact that in ℓ the size of the approximate midpoint set betweentwo points is rather small whereas it is easy to find pairs of points in L thathave infinitely many exact midpoints that are far apart. The approximatemidpoint argument has been reused many times since in various disguises.The non-embeddability result presented in this section follows from a newattempt to generalize the approximate midpoint argument of Enflo.In order to handle “unbalanced” graphs such as the parasol graphs, it isnecessary to introduce a slight generalization of the notion of δ -approximatemetric midpoints. Let ( X, d X ) be a metric space, x, y ∈ X and δ, λ ∈ (0 , λ -barycenter between x and y is any point z ∈ X satisfying d X ( x, z ) = λd X ( x, y ) and d X ( z, y ) = (1 − λ ) d X ( x, y ). Let us define the setof δ -approximate metric λ -barycenter set between x and y asBar λ ( x, y, δ ) = (cid:26) z ∈ X : max (cid:26) d X ( x, z ) λ , d X ( z, y )1 − λ (cid:27) ≤ (1 + δ ) d X ( x, y ) (cid:27) . When λ = one recovers the classical notions of metric midpoint and δ -approximate midpoint set (which will be denoted by Mid( x, y, δ ) in thesequel). In the Banach space setting, the two notions are related in thefollowing elementary way. Lemma 4.1. Let X be a Banach space. Let δ, λ ∈ (0 , . Then for every x ∈ X ,Bar λ ( − λx, (1 − λ ) x, δ ) ⊂ Mid ( − max { λ, − λ } x, max { λ, − λ } x, δ ) . Proof. Let µ := max { λ, − λ } . Assume that z ∈ Bar λ ( − λx, (1 − λ ) x, δ ).Since k − λx − (1 − λ ) x k = λ k x k one has that k z + λx k ≤ (1 + δ ) λ k x k and k z − (1 − λ ) x k ≤ (1 + δ )(1 − λ ) k x k . Consider first, the case where µ = λ . Since k λx − ( − λx ) k = 2 λ k x k , by definition of the δ -approximatemidpoint set one simply needs to show that k z + λx k ≤ (1 + δ ) λ k x k and k z − λx k ≤ (1 + δ ) λ k x k , and the former inequality holds. For the latter inequality, since λ ≥ ,(2 λ − k x k = k (1 − λ ) x − λx k≥ k z − λx k − k z − (1 − λ ) x k≥ k z − λx k − (1 + δ )(1 − λ ) k x k . Therefore, k z − λx k ≤ (2 λ − k x k + (1 + δ )(1 − λ ) k x k ≤ (1 + δ ) λ k x k .The case where µ = 1 − λ can be handled in a similar manner. Indeed,Since k (1 − λ ) x + (1 − λ ) x k = 2(1 − λ ) k x k , it remains to show that k z + (1 − λ ) x k ≤ (1 + δ )(1 − λ ) k x k and k z − (1 − λ ) x k ≤ (1 + δ )(1 − λ ) k x k , and thelatter inequality holds. For the former inequality, since λ ≤ ,(1 − λ ) k x k = k (1 − λ ) x − λx k≥ k z + (1 − λ ) x k − k z + λx k≥ k z + (1 − λ ) x k − (1 + δ ) λ k x k . Therefore, k z +(1 − λ ) x k ≤ (1 − λ ) k x k +(1+ δ ) λ k x k ≤ (1+ δ )(1 − λ ) k x k . (cid:3) It is well-known that the size of a δ -approximate metric midpoint set in anasymptotically uniformly convex Banach spaces is “small”. By “small” wemean that the set is included in the (Banach space) sum of a compact set anda ball of small radius. We shall see that δ -approximate metric λ -barycentersets in asymptotically midpoint uniformly convex Banach spaces are also“small”. Being asymptotically midpoint uniformly convexifiable is formerlyweaker than being asymptotically uniformly convexifiable. However, it isstill open whether asymptotic uniform convexity and asymptotic midpointuniform convexity are equivalent notions up to renorming. As a by-productof our work we give one more insight on this problem (cf. Section 5).Let Y be a Banach space and t ∈ (0 , δ Y ( t ) := inf y ∈ S Y sup Z ∈ cof( Y ) inf z ∈ S Z max {k y + tz k , k y − tz k} − . The norm of Y is said to be asymptotically midpoint uniformly convexif ˜ δ Y ( t ) > t ∈ (0 , S of a metric space,denoted by α ( S ), is defined as the infimum of all ε > S can becovered by a finite number of sets of diameter less than ε . Note that it is aproperty of the metric.In [11] it was shown that a Banach space Y is asymptotically midpointuniformly convex if and only iflim δ → sup y ∈ S Y α (Mid( − y, y, δ )) = 0 . To prove the main result of this section the following quantitative formula-tion of the “only if” part is needed, and the following lemma can be extractedfrom the proof of ii ) = ⇒ i ) in Theorem 2.1 from [11]. N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 37 Lemma 4.2. If the norm of a Banach space Y is asymptotically midpointuniformly convex then for every t ∈ (0 , and every y ∈ S Y one has α (cid:16) Mid (cid:16) − y, y, ˜ δ Y ( t ) / (cid:17)(cid:17) < t. The following lemma about smallness of δ -approximate metric λ -barycentersets in asymptotically midpoint uniformly convex spaces, now follows easily. Lemma 4.3. If the norm of a Banach space Y is asymptotically midpointuniformly convex then for every λ ∈ (0 , , every t ∈ (0 , and every x, y ∈ Y there exists a finite subset S of Y such thatBar λ ( x, y, ˜ δ Y ( t ) / ⊂ S + 4 t max { λ, − λ }k x − y k B Y . Proof. Let δ := ˜ δ Y ( t ) / µ := max { λ, − λ } . Assume as we may that x = y . Observe first thatBar λ ( x, y, δ ) = (1 − λ ) x + λy + Bar λ ( − λ ( y − x ) , (1 − λ )( y − x ) , δ ) . By Lemma 4.1Bar λ ( x, y, δ ) ⊆ (1 − λ ) x + λy + Mid( − µ ( y − x ) , µ ( y − x ) , δ ) ⊆ (1 − λ ) x + λy + µ k x − y k Mid (cid:18) − y − x k x − y k , y − x k x − y k , δ (cid:19) . The conclusion follows from Lemma 4.2 and the definition of the Kuratowskimeasure of non-compactness. (cid:3) We now describe a certain family G ω of sequences of graphs that con-tains the following sequences of graphs: the countably branching diamondgraphs ( D ωk ) k ∈ N , the countably branching Laakso graphs ( L ωk ) k ∈ N , and thecountably branching parasol graphs ( P ωk ) k ∈ N .A bundle with finite height is a (possibly infinite) connected graph withdistinguished nodes s and t such that all simple s - t paths have equal finitelength. The distance between the two terminals is called the height. Thesimplest of all infinite (without multiple edges) bundle with finite heightis the countably branching diamond graph of depth 1. A bundle G withfinite height has the particular property that for all vertex x ∈ G , d G ( s, x ) + d G ( x, t ) = d G ( s, t ) := h ( G ). Definition 4.1. Let G ω be the family of all sequences of graphs ( G ωk ) k ∈ N satisfying the following requirements:i) The base (directed) graph G ω is an infinite bundle with finite height.ii) G ωk +1 := G ωk ⊘ G ω , for k ≥ G ωk constructed above are infinite bundles with height h ( G ω ) k .The sequence of countably branching diamond graphs is clearly obtained bytaking the base graph to be the infinite bundle D ω , while for the countablybranching Laakso graphs and the countably branching parasol graphs thebase graphs are respectively the infinite bundles P ω and L ω depicted below.For the non-symmetric graph P ω we assume that the edges are directed fromthe bottom of the parasol to its tip. P ω bb b b b b b b b bbb L ω bbb b b b b b b b bbb Lemma 4.4. Let G ω be an infinite bundle whose terminal vertices are v b and v t . Let Y be a Banach space whose norm is asymptotically midpointuniformly convex. If f : G ω → Y is bi-Lipschitz embedding with distortion C . Then, there exists ρ := ρ ( G ω ) > such that, (56) k f ( v t ) − f ( v b ) k < Lip ( f ) (cid:18) − 15 ˜ δ Y (cid:18) ρC (cid:19)(cid:19) d G ω ( v t , v b ) . Proof. Without loss of generality assume that Lip( f ) = 1. Let h := d G ω ( v b , v t )denote the height of the bundle. Since the bundle is infinite with finiteheight by the pigeonhole principle there exists a sequence of vertices ( v i ) i ∈ N such that d ( v b , v i ) + d ( v i , v t ) = h , and k = d ( v b , v i ) for all i ∈ N . Let ρ := max { k, h − k } and λ := 1 − kh . Inequality (56) follows from the follow-ing claim. Claim 4.1. There exists j ∈ N such that (57) f ( v j ) / ∈ Bar λ (cid:18) f ( v t ) , f ( v b ) , 14 ˜ δ Y (cid:18) ρC (cid:19)(cid:19) . Indeed, assuming the claim we have either k f ( v j ) − f ( v t ) k > λ (cid:18) δ Y (cid:18) ρC (cid:19)(cid:19) k f ( v t ) − f ( v b ) k or k f ( v j ) − f ( v b ) k > (1 − λ ) (cid:18) δ Y (cid:18) ρC (cid:19)(cid:19) k f ( v t ) − f ( v b ) k , which implies in both cases k f ( v t ) − f ( v b ) k < d ( v t , v b ) (cid:18) δ X (cid:18) ρC (cid:19)(cid:19) − ≤ d ( v t , v b ) (cid:18) − 15 ˜ δ X (cid:18) ρC (cid:19)(cid:19) . N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 39 It remains to prove the claim. By Lemma 4.3 there exists a finite subset S := { s , . . . , s n } ⊂ Y such thatBar λ (cid:18) f ( v t ) , f ( v b ) , 14 ˜ δ Y (cid:18) ρC (cid:19)(cid:19) ⊂ S + 49 ρC max { λ, − λ }k f ( v t ) − f ( v b ) k B Y . If for every i , f ( v i ) ∈ Bar λ (cid:18) f ( v t ) , f ( v b ) , 14 ˜ δ Y (cid:18) ρC (cid:19)(cid:19) then f ( v i ) = s n i + y i with s n i ∈ S and y i ∈ Y so that k y i k ≤ ρC max { λ, − λ }k f ( v t ) − f ( v b ) k . However, for i = j one has k s n i − s n j k ≥ k f ( v i ) − f ( v j ) k − k y i − y j k≥ d G ω ( v i , v j ) C − ρC max { λ, − λ }k f ( v t ) − f ( v b ) k≥ C − ρC max { λ, − λ } d G ω ( v t , v b ) ≥ C − C > , which contradicts the fact that S is finite. (cid:3) The next proposition is the self-improvement argument `a la Johnson andSchechtman adapted to our setting. Proposition 4.1. Let ( G ωk ) k ∈ N ∈ G ω , and Y be an asymptotically midpointuniformly convex Banach space. There exists ρ > , such that if k ∈ N and G ωk embeds bi-Lipschitzly into Y with distortion C , then G ωk − embedsbi-Lipschitzly into Y with distortion at most C (cid:16) − ˜ δ Y ( ρC ) (cid:17) .Proof. Let Y be a Banach space with an asymptotically midpoint uniformlyconvex norm. The argument is similar to the proof of Proposition 2 . f k be a bi-Lipschitz embedding of G ωk into Y that is non-contracting and C -Lipschitz. Note that the subset of vertices V ( G ωk − ) ⊂ V ( G ωk ) := V ( G ωk − ⊘ G ω ) forms an isometric copy (up to ascaling factor s := h ( G ω )) of G ωk − . Define g k to be the restriction of theembedding f k to the subset V ( G ωk − ) ⊂ V ( G ωk ) := V ( G ωk − ⊘ G ω ) rescaledby a the factor s . Since in our setting it suffices to check the distortion onpair of adjacent vertices, it follows from Lemma 4.4 that g k is a bi-Lipschitzembedding with distortion at most C (cid:16) − ˜ δ Y ( ρC ) (cid:17) . (cid:3) Our last theorem is an asymptotic version of a result of Johnson andSchechtman in [15]. Theorem 4.1. Let ( G ωk ) k ∈ N ∈ G ω .If Y is a Banach space admitting an equivalent asymptotically midpointuniformly convex norm, then sup k ∈ N c Y ( G ωk ) = ∞ . In particular, if theequivalent asymptotically midpoint uniformly convex norm has power type p ∈ (1 , ∞ ) then c Y ( G ωk ) & k /p .Proof. Let C k := c Y ( G ωk ) and assume without loss of generality that Y isa Banach space with an asymptotically midpoint uniformly convex norm.Assume that C := sup k ≥ C k < ∞ . It follows from Proposition 4.1 and themonotonicity of the modulus that C k − ≤ C k (cid:18) − 15 ˜ δ Y (cid:18) ρC k (cid:19)(cid:19) ≤ C k (cid:18) − 15 ˜ δ Y (cid:18) ρC (cid:19)(cid:19) . Letting k go to infinity gives a contradiction.If moreover there is a constant γ > Y satisfies ˜ δ Y ( t ) ≥ γt p for some p ∈ (1 , ∞ ),then by Proposition 4.1 C k − ≤ C k (cid:18) − 15 ˜ δ Y (cid:18) ρC k (cid:19)(cid:19) ≤ C k (cid:18) − γ · (9 ρC k ) p (cid:19) . Therefore C k − C k − ≥ KC p − k , where K = γ · (9 ρ ) p , and hence C k ≥ k X j =2 KC p − j + C ≥ K ( k − C p − k . The conclusion follows easily. (cid:3) Since property ( β ) of Rolewicz implies asymptotic uniform convexity andhence asymptotic midpoint uniform convexity, Theorem 4.1 is a generaliza-tion of a result in [5] saying that if Y is a Banach space that has an equiv-alent norm with property ( β ) of Rolewicz, then sup k ∈ N c Y ( P ωk ) = ∞ andsup k ∈ N c Y ( L ωk ) = ∞ . Since there are (reflexive or not) Banach spaces thatare asymptotically uniformly convexifiable but not asymptotically uniformlysmoothable, neither the sequences ( L ωk ) k ∈ N nor ( P ωk ) k ∈ N are sequences of testspaces for the class of ( β )-renormable spaces. This gives a negative answerto Problem 4 . Applications In this last section, the main applications of our work are gathered. Thefirst application deals with the embeddability of the countably branchingdiamond graphs into certain Banach spaces. Corollary 5.1. i) There is a bi-Lipschitz embedding f k of D ωk into c +0 with distortion atmost such that for all x ∈ D ωk , | supp( f k ( x )) | ≤ k + 1 .ii) Let k ∈ N and let ε > . If p k ≥ ln(2 k +2)ln(1+ ε/ , then D ωk admits a bi-Lipschitzembedding into ℓ p k with distortion at most ε .iii) Let ( p n ) n ∈ N ⊂ (1 , ∞ ) such that lim n →∞ p n = ∞ , and let Y := ( P ∞ n =1 ℓ p n ) ℓ ,then c Y ( D ωk ) ≤ for all k ∈ N . N THE GEOMETRY OF THE COUNTABLY BRANCHING DIAMOND GRAPHS 41 Proof. i) It follows from Theorem 3.1 and Example 1.ii) It follows from i ) above that the support of f k ( x ) − f k ( y ) has size atmost 2 k + 2, and hence k f k ( x ) − f k ( y ) k p k (1 + ε/ ≤ k f k ( x ) − f k ( y ) k ∞ ≤ d k ( x, y )and d k ( x, y ) ≤ k f k ( x ) − f k ( y ) k ∞ ≤ k f k ( x ) − f k ( y ) k p k iii) it follows clearly from the previous item. (cid:3) The next corollary, where tight estimates for the L p -spaces distortion ofthe countably branching diamond graphs are obtained, complements theresults of Lee and Naor in [25] for the 2-branching diamond graphs. Corollary 5.2. i) For ≤ p < ∞ , c ℓ p ( D ωk ) ≈ k /p .ii) For < p < ∞ , c L p ( D ωk ) ≈ min { k /p , √ k } .Proof. (1) The upper estimate simply follows from Corollary 5.1 (1) andH¨older’s inequality. The lower estimate for the distortion followsfrom the fact that the modulus of asymptotic uniform convexity sat-isfies δ ℓ p ( ε ) & ε p .(2) Since L p [0 , 1] contains a linearly isometric copy of ℓ , the upper es-timate follows again from Corollary 5.1 (1) and H¨older’s inequality..For 1 < p < 2, the lower estimate was proved for D k by Lee and Naor[25]. For 2 ≤ p < ∞ , the lower estimate follows from the fact thatthe modulus of asymptotic uniform convexity satisfies δ L p ( ε ) & ε p . (cid:3) Our third application can be easily derived by combining Theorem 3.1,Theorem 3.2 and Theorem 4.1, and is the most important one. Indeed,within the class of reflexive Banach space with an unconditional asymptoticstructure it resolves the metric characterization problem in terms of graphpreclusion for the class of asymptotically uniformly convexifiable spaces.This metric characterization is only the third of this type in the asymptoticRibe program after the Baudier-Kalton-Lancien characterization [4] and theMotakis-Schlumprecht characterization [30]. Note that we can omit theseparability assumption since it follows from [10], that every (non-separable)reflexive Banach space Y with an unconditional asymptotic structure andSz( Y ∗ ) > ω has a separable subspace with the same properties. Corollary 5.3. Let Y be a reflexive Banach space with an unconditionalasymptotic structure. Then, Y ∈ h AU C i if and only if sup k ∈ N c Y ( D ωk ) = ∞ . More precisely, the sequence ( D ωk ) k ∈ N is a uniformly characterizing sequencefor the class of asymptotically uniformly convexifiable spaces within the classof reflexive Banach spaces with an unconditional asymptotic structure. The two last applications are in renorming theory. It was proven in [11],under the assumption that Y has an unconditional basis (but without as-suming reflexivity), that Y ∈ h AM U C i if and only if Y ∈ h AU C i . Whetherthis equivalence holds in full generality is still open. A simple consequenceof Theorem 4.1 and Corollary 5.3 is that the equivalence holds for separablereflexive Banach space with an unconditional asymptotically structure. Corollary 5.4. Let Y be a separable reflexive Banach space with an uncon-ditional asymptotic structure. Then Y ∈ h AM U C i if and only if Y ∈ h AU C i . Also, as a consequence of Corollary 3.1 and the fact that the 2-branchingdiamond graphs form a sequence of test-spaces for the class uniformly con-vexifiable Banach spaces [15] one obtains: Corollary 5.5. let Y be a Banach space and let < p < ∞ . Then, L p ([0 , , Y ) ∈ h AM U C i if and only if Y ∈ h U C i . It is worth noticing that there exist reflexive Banach spaces of type 2 thatare not super-reflexive ([14], [40]). Let Y be such a space, it follows that L ( Y ) has type 2 but sup k ∈ N c L ( Y ) ( D ωk ) < ∞ . Acknowledgments: The authors wish to thank the organizers of theWorkshop in Analysis and Probability, W. B. Johnson, D. Kerr, and G.Pisier. Most of this work was carried out when the third and fourth authorswhere participants of the workshop in July 2015 and July 2016. References [1] S. A. Argyros and I. Deliyanni, Examples of asymptotic l Banach spaces , Trans. Amer.Math. Soc. (1997), no. 3, 973–995.[2] K. Ball, The Ribe programme , Ast´erisque (2013), Exp. 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E-mail address : [email protected] Thomas Schlumprecht, Department of Mathematics, Texas A&M University, CollegeStation, TX 77843-3368, USA, and Faculty of Electrical Engineering, Czech TechnicalUniversity in Prague, Zikova 4, 16627, Prague, Czech Republic E-mail address : [email protected] Sheng Zhang, School of Mathematics, Southwest Jiaotong University, Chengdu,Sichuan 611756, China E-mail address ::