Test-space characterizations of some classes of Banach spaces
aa r X i v : . [ m a t h . F A ] D ec Test-space characterizations of some classes of Banach spaces
M. I. OstrovskiiNovember 5, 2018
Abstract.
Let P be a class of Banach spaces and let T = { T α } α ∈ A be a set of metric spaces. Wesay that T is a set of test-spaces for P if the following two conditions are equivalent: (1) X / ∈ P ; (2) The spaces { T α } α ∈ A admit uniformly bilipschitz embeddings into X .The first part of the paper is devoted to a simplification of the proof of the following test-spacecharacterization obtained in M. I. Ostrovskii [Different forms of metric characterizations of classesof Banach spaces, Houston J. Math. , to appear]:For each sequence { X m } ∞ m =1 of finite-dimensional Banach spaces there is a sequence { H n } ∞ n =1 of finite connected unweighted graphs with maximum degree 3 such that the following conditionson a Banach space Y are equivalent: (A) Y admits uniformly isomorphic embeddings of { X m } ∞ m =1 ; (B) Y admits uniformly bilipschitz embeddings of { H n } ∞ n =1 .The second part of the paper is devoted to the case when { X m } ∞ m =1 is an increasing sequenceof spaces. It is shown that in this case the class of spaces given by (A) can be characterized usingone test-space, which can be chosen to be an infinite graph with maximum degree 3. Primary: 46B07; Secondary: 05C12, 46B85, 54E35
Contents
155 References 24
Embeddings of metric spaces into Banach spaces play an important role in ComputerScience (see, for example, [WS11, Chapter 15]) and in Topology (see [Yu06]). Inconnection with problems of embeddability of metric spaces into Banach spaces it ould be interesting to find metric characterizations of well-known classes of Banachspaces. By metric characterizations we mean characterizations which can be testedon an arbitrary metric space. So, in a metric characterization only distances betweenelements of the space are involved, and no linear combinations of any kind can beused. At this point it seems to be unclear: how to define the notion of a metriccharacterization which would be the most useful for applications in the theory ofmetric embeddings? One can try to define a metric characterization in the followingway: a metric characterization is a set of formulas understood as in logic (see [Han77,p. 19] for a definition of a first-order formula). Some of the variables in the formulasare elements in an unknown metric space X (so the formulas make sense for anarbitrary metric space X ). We say that such set of formulas characterizes a class P of Banach spaces if the following conditions are equivalent for a Banach space X : • X ∈ P• All of the formulas of the set hold for X .Metric characterizations which we are going to study in this paper use the fol-lowing definition. Definition 1.1.
Let 0 ≤ C < ∞ . A map f : ( A, d A ) → ( Y, d Y ) between two metricspaces is called C - Lipschitz if ∀ u, v ∈ A d Y ( f ( u ) , f ( v )) ≤ Cd A ( u, v ) . A map f is called Lipschitz if it is C -Lipschitz for some 0 ≤ C < ∞ . For a Lipschitzmap f we define its Lipschitz constant byLip f := sup d A ( u,v ) =0 d Y ( f ( u ) , f ( v )) d A ( u, v ) . Let 1 ≤ C < ∞ . A map f : A → Y is called a C -bilipschitz embedding if thereexists r > ∀ u, v ∈ A rd A ( u, v ) ≤ d Y ( f ( u ) , f ( v )) ≤ rCd A ( u, v ) . (1)A bilipschitz embedding is an embedding which is C -bilipschitz for some 1 ≤ C < ∞ .The smallest constant C for which there exist r > distortion of f . A set of embeddings is called uniformly bilipschitz if they haveuniformly bounded distortions. Remark . Linear embeddings T i : A i → Y of Banach (or normed) spaces into aBanach (normed) space are uniformly bilipschitz if and only ifsup i (cid:0) || T i || · || T − i | T i ( A i ) || (cid:1) < ∞ . Such embeddings T i are called uniformly isomorphic . emark . The definition of a metric characterization suggested above does notseem to be completely satisfactory. It includes trivial (in a certain sense) charac-terizations of the type: A Banach space is nonreflexive if and only if it containsa (separable) subset which is bilipschitz equivalent to a nonreflexive separable Ba-nach space. (The validity of this characterization is a consequence of the followingwell-known facts: (1) Each nonreflexive Banach space contains a separable nonreflex-ive subspace. This fact follows, for example, from the Eberlein–Shmulian theorem[DS58, Theorem V.6.1]; (2) If a Banach space Y admits a bilipschitz embeddingof a (separable) nonreflexive Banach space X , then Y is nonreflexive, see [HM82,Lemma 3.1] or [BL00, Theorem 7.9].)At this point it is not clear how to define a metric characterization in such away that, on one hand, all interesting examples are included, but, on the otherhand, characterizations like the trivial characterization of nonreflexivity mentionedin Remark 1.3 are excluded. We shall focus on one of the classes of metric charac-terizations which is known to be interesting (see [Bau07, Bau09+, Bou86, BMW86,JS09, MN08, Ost11a, Ost11d, Pis86]). We mean metric characterizations of thefollowing type. Definition 1.4.
Let P be a class of Banach spaces and let T = { T α } α ∈ A be a setof metric spaces. We say that T is a set of test-spaces for P if the following twoconditions are equivalent1. X / ∈ P .2. The spaces { T α } α ∈ A admit uniformly bilipschitz embeddings into X . Remark . We use
X / ∈ P in condition 1 of Definition 1.4 rather than X ∈ P forterminological reasons: we would like to use terms “test-spaces for reflexivity, su-perreflexivity, etc.” rather than “test-spaces for non reflexivity, non superreflexivity,etc.” Remark . One can introduce the notion of test-spaces differently, requiring, forexample: “at least one of the spaces { T α } α ∈ A admits a bilipschitz embedding into X ”. However, this version of test-space characterizations includes the trivial char-acterization of reflexivity mentioned in Remark 1.3. Another reason why we havechosen the introduced in Definition 1.4 notion of test-spaces is: many importantknown characterizations are of this form (see [Bau07, Bau09+, Bou86, BMW86,JS09, MN08, Ost11a, Ost11d, Pis86]). The main purpose of this section is to give a simpler proof of the following result of[Ost11d]: heorem 2.1 ([Ost11d]) . For each sequence { X m } ∞ m =1 of finite-dimensional Banachspaces there exists a sequence { H n } ∞ n =1 of finite connected unweighted graphs withmaximum degree such that the following conditions on a Banach space Y areequivalent: • Y admits uniformly isomorphic embeddings of { X m } ∞ m =1 . • Y admits uniformly bilipschitz embeddings of { H n } ∞ n =1 . Everywhere in this paper we consider graphs as metric spaces with their shortestpath metric : the distance between two vertices is the length of the shortest pathbetween them. In some cases we consider weighted graphs with some positive weightsassigned to their edges. In such a case the length of the path is the sum of weightsof edges included in it. For graphs with no weights (sometimes we emphasize thisby calling them unweighted graphs ) the length of a path is the number of edges init (this corresponds to the case when all weights are equal to 1).
Remark . For the reasons explained in [Ost11d] we restrict our attention to thecase sup m dim X m = ∞ .Our purpose is to simplify the proof of the step which has the longest proof in[Ost11d]. We shall also present other steps of the proof, in a more general form thanin [Ost11d]. The reason for doing so is that we need these steps in the more generalform later in the paper. Recall some standard definitions. Definition 2.3.
Let α >
0. We say that a subset A in a metric space ( X, d ) is α -dense in a subset B ⊂ X if A ⊂ B and ∀ x ∈ B ∃ y ∈ A d ( x, y ) ≤ α. A subset D in a metric space ( X, d ) is called α -separated if d ( x, y ) ≥ α for each x, y ∈ D , x = y .If A and B are subsets in a metric space ( X, d ), we let d ( A, B ) = inf { d ( x, y ) : x ∈ A, y ∈ B } . First we introduce an approximate description of convex sets in Banach spacesusing unweighted graphs. Let C be a convex set in a Banach space X , α, β > V be an α -separated β -dense subset of C . Remark . It is easy to see that such subset V does not exist if β < α anddiam C > β . In this paper only the case where β ≥ α is considered. In this case theexistence is immediate from Zorn’s lemma. Definition 2.5.
Let G be the graph whose vertex set is V and whose edge set isdefined in the following way: vertices u, v ∈ V are joined by an edge if and onlyif || u − v || ≤ β . The graph G is called a ( C, α, β ) -graph . If α = β , G is called a( C, α ) -graph . t is easy to check that G is not uniquely determined by C , α , and β , but for ourpurposes it does not matter which of ( C, α, β )-graphs we pick. We endow the vertexset V of G with its shortest path metric d G . Lemma 2.6.
The natural embedding f : ( V, d G ) → ( X, || · || ) is bilipschitz withdistortion ≤ max (cid:8) , βα (cid:9) . More precisely, Lip( f ) ≤ β and Lip( f − | f ( V ) ) ≤ max (cid:26) β , α (cid:27) . Proof.
The inequality Lip( f ) ≤ β follows immediately from the fact that adjacentvertices in G are at distance ≤ β in X , and the definition of the shortest pathmetric.To prove the inequality for Lip( f − ) we consider two distinct vertices u, v ∈ V ,write || u − v || = dβ for some d >
0, and consider two cases:
Case A. d ≤
3. In such a case d G ( u, v ) = 1. Since || u − v || ≥ α , we have d G ( u, v ) || u − v || ≤ α . Case B. d >
3. In this case, and even in a wider case d >
2, we show that d G ( u, v ) ≤ ⌊ d ⌋ − d G ( u, v ) || u − v || ≤ ⌊ d ⌋ − dβ ≤ β . We prove the inequality (2) by induction starting with 2 < d ≤
3. In this case d G ( u, v ) = 1 and so it is clear that (2) is satisfied.Suppose that we have proved the inequality (2) for 2 < d ≤ n . Let us show thatthis implies the inequality for n < d ≤ n + 1. We do this as follows:Consider the vertex e u lying on the line segment joining u and v at distance 2 β from u . Since e u ∈ C (this is the point where we use the convexity of C ), there is w ∈ V satisfying || w − e u || ≤ β .By the triangle inequality, we have || w − u || ≤ β and || w − v || ≤ ( d − β .The first inequality implies d G ( w, u ) = 1. Applying the triangle inequality and theInduction Hypothesis, we get d G ( u, v ) ≤ d G ( w, v ) + 1 ≤ ( ⌊ d − ⌋ −
1) + 1 = ⌊ d ⌋ − || w − v || ≤ β , but in this case theresult is also easy to verify.)If X is a Banach space, we use the notation B X ( r ), r >
0, for { x ∈ X : || x || ≤ r } .The unit ball B X (1) is also denoted by B X . Observe that if X is finite-dimensional(and β ≥ α ), ( B X ( r ) , α, β )-graphs are finite.The next step in the proof of Theorem 2.1 in [Ost11d] is the following lemma (in[Ost11d] the lemma is stated in slightly less general form). emma 2.7. If { X m } ∞ m =1 are finite-dimensional Banach spaces, and a Banach space Y admits uniformly bilipschitz embeddings of a collection of ( B X m ( n ) , α ( n ) , β ( n )) -graphs m, n ∈ N , where α ( n ) ≤ β ( n ) , lim n →∞ α ( n ) = 0 , lim n →∞ β ( n ) = 0 , and sup n β ( n ) α ( n ) < ∞ , then { X m } ∞ m =1 are uniformly isomorphic to subspaces of Y . Lemma 2.7 can be derived from the following discretization result.
Theorem 2.8.
For each finite-dimensional Banach space X and each γ > thereexists ε > such that for each bilipschitz embedding L of an ε -dense subset of B X ,with the metric inherited from X , into a Banach space Y there is a linear embedding T : X → Y such that (1 − γ ) || T || · || T − || ≤ ( the distortion of L ) . This theorem goes back to Ribe [Rib76], a new proof of the essential ingredientswas found by Heinrich-Mankiewicz [HM82], versions of these proofs are presentedin [BL00]. The first explicit bound on ε was found by Bourgain [Bou87]. Bourgain’sproof was simplified and explained by Begun [Beg99] and Giladi-Naor-Schechtman[GNS11]. We recommend everyone who would like to study this result to start byreading [GNS11].It is clear that the fact that the ball in Theorem 2.8 has radius 1 plays norole. To derive Lemma 2.7 from Theorem 2.8 we observe that the vertex set ofa ( B X m ( n ) , α ( n ) , β ( n ))-graph is a β ( n )-dense subset of B X m ( n ). Furthermore, byLemma 2.6, the graph distance on this set is max n , β ( n ) α ( n ) o -equivalent to the metricinherited from X . This proves Lemma 2.7.Lemmas 2.6 and 2.7 show that a Banach space Y admits uniformly isomorphicembeddings of X m if and only if Y admits uniformly bilipschitz embeddings of some(or any) collection of ( B X m ( n ) , α ( n ) , β ( n ))-graphs m, n ∈ N , where α ( n ) ≤ β ( n ),lim n →∞ α ( n ) = 0, lim n →∞ β ( n ) = 0, and sup n β ( n ) α ( n ) < ∞ . This statement does notcomplete the proof of Theorem 2.1 (even of a weaker version of it, with 3 replacedby any other uniform bound on degrees of { H n } ∞ n =1 ). In fact, it is easy to seethat if sup m dim X m = ∞ , the degrees of any collection of ( B X m ( n ) , α ( n ) , β ( n ))-graphs, m, n ∈ N , where α ( n ) ≤ β ( n ), lim n →∞ α ( n ) = 0, lim n →∞ β ( n ) = 0, andsup n β ( n ) α ( n ) < ∞ , are unbounded. Observation 2.9.
Since we assumed sup m dim X m = ∞ , each Banach space ad-mitting uniformly isomorphic embeddings of X m has to be infinite-dimensional. Forthis reason, in order to prove Theorem 2.1 it suffices to show that for each finite-dimensional Banach space X , each ( B X ( r ) , δ ) -graph G (0 < δ < r < ∞ ) , and eachinfinite-dimensional Banach space Z containing X as a subspace, there exist a graph H and bilipschitz embeddings ψ : G → H and ϕ : H → Z , such that distortions of ψ and ϕ are bounded from above by absolute constants and the maximum degree of H is . It is clear that it is enough to consider the case δ = 1 .Remark . Observation 2.9 is the main step towards simplification of the proofof Theorem 2.1 given in [Ost11d]: in [Ost11d] the graph H was embedded into X if dim X ≥ H which we use is the same as in [Ost11d]: We introducethe graph M G as the following “expansion” of G : we replace each edge in G by apath of length M . It is clear that the graph M G is well-defined for each M ∈ N .In our construction of H the number M will be chosen to be much larger than thenumber of edges of G . We use the term long paths for the paths of length M whichreplace edges of G . Next step in the construction of H : For each vertex v of G , weintroduce a path p v in the graph H whose length is equal to the number of edges of G , we call each such path a short path . At the moment these paths do not interact.We continue our construction of H in the following way. We label vertices of shortpaths in a monotone way by long paths. “In a monotone way” means that the firstvertex of each short path corresponds to the long path p , the second vertex of each short path corresponds to the long path p etc. We complete our construction of H introducing, for a long path p in M G corresponding to an edge uv in G , a path ofthe same length in H (we also call it long ) which joins those vertices of the shortpaths p u and p v which have label p . There is no further interaction between shortand long paths in H . It is obvious that the maximum degree of H is 3.It remains to define embeddings ψ and ϕ and to estimate their distortions.To define ψ we pick a long path p in M G (in an arbitrary way) and map eachvertex u of G onto the vertex in H having label p in the short path p u correspondingto u . The estimates for Lip( ψ ) and Lip( ψ − ) given below are taken from [Ost11d].We reproduce them because they do not take much space.We have Lip( ψ ) ≤ e ( G ) + M , where e ( G ) is the number of edges of G . Infact, to estimate the Lipschitz constant it suffices to find an estimate from above forthe distances in H between ψ ( u ) and ψ ( v ) where u and v are adjacent vertices of G . To see that 2 e ( G ) + M provides the desired estimate we consider the followingthree-stage walk from ψ ( u ) from ψ ( v ): • We walk from ψ ( u ) along the short path p u to the vertex labelled by the longpath corresponding to the edge uv in G . • Then we walk along the corresponding long path to its end in p v . • We conclude the walk with the piece of the short path p v which we need totraverse in order to reach ψ ( v ).We claim that Lip( ψ − ) ≤ M − . This gives an absolute upper bound for thedistortion of ψ provided the quantity e ( G ) is controlled by M , we need M to bemuch larger than e ( G ) only if we would like to make the distortion close to 1. Toprove Lip( ψ − ) ≤ M − we let ψ ( u ) and ψ ( v ) be two vertices of ψ ( V ( G )). We needto estimate d G ( u, v ) from below in terms of d H ( ψ ( u ) , ψ ( v )). Let P = ψ ( u ) , w , . . . , w n − , ψ ( v ) e one of the shortest ψ ( u ) ψ ( v )-paths in H . Let u, u , . . . , u k − , v be those verticesof G for which the path P visits the corresponding short paths p u , p u , . . . , p u k − , p v .We list u , . . . , u k − in the order of visits. It is clear that in such a case the sequence u, u , . . . , u k − , v is a uv -walk in G . Therefore d G ( u, v ) ≤ k . On the other hand, in H , to move from one short path to another, one has to traverse at least M edges,therefore d H ( ψ ( u ) , ψ ( v )) ≥ kM . This implies Lip( ψ − ) ≤ M − .Our next purpose is to introduce ϕ : H → Z . First we prove (Lemma 2.11 below)that there is a bilipschitz embedding of M G into some finite-dimensional subspace W of Z with distortion bounded by an absolute constant.It is convenient to handle all M ∈ N simultaneously by considering the followingthickening of the graph G (see [Gro93, Section 1.B] for the general notion of thick-ening). For each edge uv in G we join u and v with a set isometric to [0 , t ( uv ). The thickening T G of G is the union of all sets t ( uv ) (suchsets can intersect at their ends only) with the distance between two points definedas the length of the shortest curve joining the points. Lemma 2.11.
If a finite unweighted graph G endowed with its graph distance admitsa bilipschitz embedding τ into a finite-dimensional Banach space X , then the graph T G admits a bilipschitz embedding f into any infinite-dimensional Banach space Z containing X as a subspace, and the distortion of f is bounded in terms of thedistortion of τ and some absolute constants.Remark . In [Ost11d, Section 4] a stronger result was proved, namely, it wasproved that in the case where dim X ≥
3, the bilipschitz embedding f whose exis-tence is claimed in Lemma 2.11 can be required to map T G into X . As we shallsee, this result is not needed for Theorem 2.1. However, it could be of independentinterest. Proof of Lemma 2.11.
We may assume without loss of generality that Lip( τ − ) = 1,that is, || τ ( u ) − τ ( v ) || ≥ d G ( u, v ). We construct a bilipschitz embedding of T G into an (arbitrary) Banach space W containing X as a subspace and satisfyingdim( W/X ) = e ( G ), where e ( G ) is the number of edges in G and W/X is thequotient space.We find an Auerbach basis in
W/X . Recall the definition. Let { x i } ni =1 be abasis in an n -dimensional Banach space Y , its biorthogonal functionals are definedby x ∗ i ( x j ) = δ ij (Kronecker delta). The basis { x i } ni =1 is called an Auerbach basis if || x i || = || x ∗ i || = 1 for all i ∈ { , . . . , n } . This notion goes back to [Aue30]. See[Ost11b, Section 2] and [Pli95] for historical comments and proofs.Since the cardinality of the Auerbach basis is equal to the number of edges in G , we label its elements by edges. Also we lift the elements of this Auerbach basisinto W . Since W is finite-dimensional, we may assume that the norms of the liftedelements are also equal to 1. We use the notation { e uv } uv ∈ E ( G ) for the lifted elements f the Auerbach basis and the notation { e ∗ uv } uv ∈ E ( G ) for its biorthogonal system. Itis clear that { e ∗ uv } uv ∈ E ( G ) may be regarded as elements of W ∗ .If uv is an edge in G , we map t ( uv ) onto the concatenation of two line seg-ments in W ⊂ Z , namely, onto the concatenation of h τ ( u ) , τ ( u )+ τ ( v )2 + e uv i and h τ ( u )+ τ ( v )2 + e uv , τ ( v ) i . More precisely, we map the point in t ( uv ) which is at dis-tance α ∈ [0 ,
1] from u onto the point ατ ( v ) + (1 − α ) τ ( u ) + min { α, − α ) } e uv . (3)It is easy to check that such points cover the concatenation of the line segments h τ ( u ) , τ ( u )+ τ ( v )2 + e uv i and h τ ( u )+ τ ( v )2 + e uv , τ ( v ) i . We denote this map by f : T G → W .We claim that f is a bilipschitz embedding and its distortion is bounded in termsof distortion of τ and some absolute constant. To estimate Lip( f ) observe that thederivative of the function in (3) with respect to α is τ ( v ) − τ ( u ) ± e uv (at pointswhere the derivative is defined). Hence Lip( f ) ≤ Lip( τ ) + 2.To estimate the Lipschitz constant of f − , we need to estimate from above thequotient d T G ( x, y ) || f ( x ) − f ( y ) || , where x, y ∈ T G . Let α be the distance in T G from x ∈ t ( uv ) to u and let β bethe distance from y ∈ t ( wz ) to w . We may choose our notation in such a way that α, β ≤ . Let D = d T G ( x, y ).First we consider the case when the edges uv and wz are different. We have d T G ( u, w ) ≥ D − α − β . Thus || f ( u ) − f ( w ) || = || τ ( u ) − τ ( w ) || ≥ D − α − β and || f ( x ) − f ( y ) || ≥ D − (Lip( τ ) + 3)( α + β ). Here we use the fact that, sinceLip( f ) ≤ Lip( τ ) + 2, we have || f ( x ) − f ( u ) || ≤ (Lip( τ ) + 2) α and || f ( y ) − f ( w ) || ≤ (Lip( τ ) + 2) β . On the other hand || f ( x ) − f ( y ) || ≥ e ∗ uv ( f ( x ) − f ( y )) = e ∗ uv ( f ( x ) − f ( u )) + e ∗ uv ( f ( u ) − f ( y )) = e ∗ uv ( f ( x ) − f ( u )) = 2 α (we use uv = wz , the definitionof α and the fact that { e uv } is a lifted Auerbach basis in W/X ). Similarly we get || f ( x ) − f ( y ) || ≥ β . Therefore d T G ( x, y ) || f ( x ) − f ( y ) || ≤ min (cid:26) D max { , D − (Lip( τ ) + 3)( α + β ) } , D α , D β (cid:27) . It is easy to see that the minimum in this inequality is bounded from above in termsof Lip( τ ) and an absolute constant. If α = 0, or β = 0, or both, we modify thisargument in a straightforward way.It remains to consider the case where x, y ∈ t ( uv ). Let d T G ( x, u ) = α and T G ( y, u ) = β , so d T G ( x, y ) = | α − β | . It is easy to see that | e ∗ uv ( f ( x ) − f ( y )) | = ( | α − β | if α and β are on the same side of | − α − β | otherwise.In the former case we get d T G ( x, y ) || f ( x ) − f ( y ) || ≤ . If the latter case we use f ( x ) − f ( y ) = ( α − β )( τ ( v ) − τ ( u )) ± − α − β ) e uv . We get from here that || f ( x ) − f ( y ) || ≥ max { | − α − β | , | α − β ||| τ ( u ) − τ ( v ) || − | − α − β |} . The desired estimate follows.This proof shows that there exists an embedding ϕ : M G → W such thatLip( ϕ ) ≤ ϕ − ) is bounded from above by an absolute constant. The restof the proof is quite similar to the proof in [Ost11d], we only replace the embedding ϕ : M G → X constructed in [Ost11d] by the embedding ϕ : M G → W constructedin Lemma 2.11. For convenience of the reader we reproduce this part of the proofwith necessary modifications. We number vertices along short paths using numbersfrom 1 to e ( G ) in such a way that vertices numbered 1 correspond to the same longpath in the correspondence described above.At this point we are ready to describe the action of the map ϕ on vertices ofshort paths. We construct the map ϕ as a map into the Banach space W ⊕ R .This is enough because W ⊕ R admits a linear bilipschitz embedding into Z withdistortion bounded by an absolute constant.For vertex w of H having number i on the short path p u the image of w in W ⊕ R is ϕ ( w ) = ϕ ( u ) ⊕ i (here we use the same notation u both for a vertex of G andthe corresponding vertex in M G ).To map vertices of long paths of H into W ⊕ R we observe that the numberingof vertices of short paths leads to a one-to-one correspondence between long pathsand numbers { , . . . , e ( G ) } . We define the map ϕ on a long path corresponding to i by ϕ ( w ) = ϕ ( w ′ ) ⊕ i , where w ′ is the uniquely determined vertex in a long pathof M G corresponding to a vertex w in a long path of H .The fact that Lip( ϕ ) ≤ ϕ -images of adjacent vertices of H is at most 1 (here we useLip( ϕ ) ≤ ϕ − ). In this part of the proof we assume that M > e ( G ). Let w and z be two vertices of H . As we have already mentioned our onstruction implies that there are uniquely determined corresponding vertices w ′ and z ′ in M G .Obviously there are two possibilities:(1) d MG ( w ′ , z ′ ) ≥ d H ( w, z ). In this case we observe that the definitions of ϕ and of the norm on W ⊕ R imply that || ϕ ( w ) − ϕ ( z ) || ≥ || ϕ ( w ′ ) − ϕ ( z ′ ) || ≥ d MG ( w ′ , z ′ ) / Lip( ϕ − ) ≥ d H ( w, z ) / Lip( ϕ − ) . (2) d MG ( w ′ , z ′ ) < d H ( w, z ). This inequality implies that there is a path joining w and z for which the naturally defined short-paths-portion is longer than the long-paths-portion . The inequality M > e ( G ) implies that the short-paths-portion of thispath consists of one path of length > d H ( w, z ). This implies that the differencebetween the second coordinates of w and z in the decomposition W ⊕ R is > d H ( w, z ). Thus || ϕ ( w ) − ϕ ( z ) || > d H ( w, z ).Since Lip( ϕ − ) ≥ ϕ ) ≤ ϕ − ) ≤ ϕ − ) in each of the cases (1) and (2). The proof of Theorem 2.1 iscompleted. Bourgain [Bou86] proved that a Banach space is nonsuperreflexive if and only if it ad-mits uniformly bilipschitz embeddings of binary trees of all finite depths (see [BL00,pp. 412, 436] for the definition and equivalent characterizations of superreflexivity).Baudier [Bau07] strengthened the “only if” part of this result by proving that eachnonsuperreflexive Banach space admits a bilipschitz embedding of an infinite binarytree.Our purpose in this section is to find similar one-test-space-characterizationsfor classes of Banach spaces defined in terms of excluded finite-dimensional sub-spaces. At this moment we do not know how to do this for an arbitrary sequenceof finite-dimensional subspaces, we found such a characterization only for increasingsequences of finite-dimensional subspaces.
Theorem 3.1.
Let { X n } ∞ n =1 be an increasing sequence of finite-dimensional Banachspaces with dimensions going to ∞ . Then there exists an infinite graph G such thatthe following conditions are equivalent: • G admits a bilipschitz embedding into a Banach space X . • The spaces { X n } admit uniformly isomorphic embeddings into X .Remark . Theorem 3.1 is obviously weaker that Theorem 4.1 proved in Section4. We give an independent proof of Theorem 3.1 because it is substantially simpler. roof of Theorem 3.1. Our proof is based on the construction of graphs providingapproximate descriptions of convex sets, see Definition 2.5. We use the followingimmediate consequence of Theorem 2.8:
Lemma 3.3.
Let X be a finite-dimensional Banach space, Y be a Banach space ad-mitting uniformly bilipschitz embeddings of some ( B X ( n ) , -graphs, and C ∈ [1 , ∞ ) be an upper bound for distortions of these embeddings. Then for each ε > there isa linear embedding T : X → Y satisfying || T || · || T − || ≤ ε ) C . Let L be the inductive limit of the sequence { X n } , that is, L = S ∞ n =1 X n withits natural vector operations and the norm whose restriction to each of X n is thenorm of X n . So L is an incomplete normed space (we can, of course, consider itscompletion, but for our purposes completeness is not needed). We construct G asa graph whose vertex set V is a countable infinite subset of L ⊕ R . (We fix the ℓ -sum because it is convenient to have a precise formula for the norm, of course inthe bilipschitz category all direct sums are equivalent.) The main features of theconstruction are:(1) The graph G with its shortest path metric is a locally finite metric space.(Recall that a metric space is called locally finite if all balls of finite radius init have finite cardinality.)(2) We have V = S ∞ n =1 V n , where V n are finite and there exist uniformly bilipschitzembeddings f n : V n → ( X n ⊕ R ), where we assume that the distance in V n isinherited from G .(3) The set V endowed with its shortest path metric d G contains images of bilips-chitz embeddings of some ( B X n ( m ) , , m, n ∈ N , with uniformly bounded distortions.First let us explain why such graph G satisfies the conclusion of Theorem 3.1.Suppose that X is such that the spaces { X n } admit uniformly isomorphic em-beddings into X . Then condition (2) implies that V n admit uniformly bilipschitzembeddings into X . Since G is locally finite, by the main result of [Ost11c], we getthat G admits a bilipschitz embedding into X .Now suppose that G admits a bilipschitz embedding into X . By (3) we get that X admits uniformly bilipschitz embeddings of some ( B X n ( m ) , m, n ∈ N .Applying Lemma 3.3, we get that the spaces { X n } admit uniformly isomorphicembeddings into X .We construct V as an infinite union S ∞ n =1 V n , where each V n is a finite subset of C n := conv n [ k =1 ( B X k ( k ) , s k ) ! ⊂ L ⊕ R , where s k ∈ R , the pairs ( z, s k ) are in the sense of the decomposition L ⊕ R , and( B X k ( k ) , s k ) = { ( z, s k ) : z ∈ B X k ( k ) } . Recall that B X k ( k ) is the centered at 0 ballof X k of radius k . ow we describe our choice of V n and s n such that the conditions (1)–(3) aboveare satisfied. One of the requirements is s n +1 − s n > . (4)We let s = 0 and let V to be a 1-separated 1-dense subset of ( B X (1) , s ) (seeDefinition 2.3).The choice of s is less restrictive than further choices. We let s = 2 and let V be the extension of V to a 1-separated 1-dense subset of C := conv [ n =1 ( B X n ( n ) , s n ) ! satisfying the condition: some part of V is a 1-dense in ( B X (2) , s ), and some partof it is a 1-dense in ( B X (2) , s ) (we use s − s > A of L ⊕ R : A [ a, b ] := A ∩ ( L ⊕ [ a, b ]) , (5)where [ a, b ] is an interval in R . We use similar notation for open and half-openintervals.Now we turn to the choice of s . We choose s to satisfy (4) and to be so largethat V is (cid:0) (cid:1) -dense in C [ s , s ]. It is easy to see that sufficiently large s satisfythese conditions. Then we extend V to a 1-separated (cid:0) (cid:1) -dense in C subset V in such a way that • ( V \ V ) ⊂ C ( s , s ] • Some parts of V are (cid:0) (cid:1) -dense in the sets ( B X (3) , s ), ( B X (3) , s ), and( B X (3) , s ), respectively (here we use (4)).We continue in the following way. We pick s in such a way that • V is (cid:0) + (cid:1) -dense in C [ s , s ]. • V is (cid:0) + (cid:1) -dense in C [ s , s ].Now we extend V to a 1-separated (cid:0) + (cid:1) -dense subset V of C in such away that • ( V \ V ) ⊂ C ( s , s ] • Some parts of V are (cid:0) + (cid:1) -dense subsets in ( B X (4) , s ), ( B X (4) , s ),( B X (4) , s ), and ( B X (4) , s ), respectively (here we use (4)). e continue in an obvious way: In step n we pick s n , s n − s n − >
1, in such away that for each m = 2 , . . . , n − V m is a (cid:16) + + · · · + (cid:0) (cid:1) n − (cid:17) -dense subset of C n [ s , s m ].We extend V n − to a 1-separated (cid:16) + · · · + (cid:0) (cid:1) n − (cid:17) -dense subset V n of C n in such a way that • ( V n \ V n − ) ⊂ C n ( s n − , s n ] • Some parts of V n are (cid:16) + · · · + (cid:0) (cid:1) n − (cid:17) -dense subsets in( B X ( n ) , s n ) , . . . , ( B X n ( n ) , s n ) , respectively (here we use (4)).Let V = S ∞ n =1 V n and C = conv ∞ [ n =1 ( B X n ( n ) , s n ) ! ⊂ L ⊕ R . Our construction implies that V is a 1-separated 2-dense subset of C . We let G be the corresponding ( C, , G satisfies the conditions (1)–(3) above.Condition (1). The set V is a locally finite subset of L ⊕ R because it is containedin L ⊕ [ s , ∞ ) and its intersection with each subset of the form L ⊕ [ s , s n ] is a finiteset V n . The graph G is locally finite because, by Lemma 2.6, its natural embeddinginto L ⊕ R is bilipschitz.Condition (2). We apply Lemma 2.6 to V , the corresponding ( C, , L ⊕ R . We get that the natural embedding of V with the metric inherited from G into L ⊕ R is bilipschitz. Hence its restrictions to V n are uniformly bilipschitz. Thefact that the restriction of this map to V n maps V n into X n ⊕ R follows from thedefinitions.Condition (3). Our construction of set V is such that it contains subsets whichare 1-separated 2-dense subsets in shifted B X m ( n ). We apply Lemma 2.6 twice. Firsttime to V , the corresponding ( C, , L ⊕ R . Second time we apply itto the set ( B X m ( n ) , s n ), the corresponding (( B X m ( n ) , s n ) , , V with ( B X m ( n ) , s n )). We get that embeddings ofall of these graphs into L ⊕ R have uniformly bounded distortions. Therefore themetrics of these (( B X m ( n ) , s n ) , , G . Hence the condition (3) is also satisfied. Characterization in terms of an infinite graph with max-imum degree Our next purpose is to show that the test-space for Theorem 3.1 can be chosen tohave maximum degree 3:
Theorem 4.1.
Let { X n } ∞ n =1 be an increasing sequence of finite-dimensional Banachspaces with dimensions going to ∞ . Then there exists an infinite graph H withmaximum degree such that the following conditions are equivalent: • H admits a bilipschitz embedding into a Banach space X . • The spaces { X n } admit uniformly isomorphic embeddings into X .Proof. Our proof uses some of the ideas of the proof of Theorem 3.1. For this reasonwe keep the same notation for some of the objects, although now they are somewhatdifferent. We use Definition 2.3 and the notation introduced in formula (5).We may assume without loss of generality that dim X n = n . We introduce convexsets C n and C in L ⊕ R of the form C n := conv n [ k =1 ( B X i ( k ) (4 k ) , s k ) ! and C = ∞ [ n =1 C n , where { i ( k ) } ∞ k =1 is a sequence of natural numbers satisfying i (1) = 1, i ( k ) ≤ i ( k +1) ≤ i ( k ) + 1, and such that the equality i ( k + 1) = i ( k ) + 1 holds rarely (the exactcondition will be described later), so the dimension of the sets C n increases slowly.Let { s n } ∞ n =1 be a sequence of real numbers, such that s = 0 and the followingtwo conditions are satisfied: Gap condition: s n − s n − > n +1 . (6) Density condition: C n − [ a, s n − ] is (cid:18) (cid:19) n − − dense in C n [ a, s n − ] for every a ∈ [ s , s n − ] . (7) Remark . It is easy to verify that condition (7) is satisfied for each sufficientlyrapidly increasing sequence { s n } ∞ n =1 .We construct two sequences of finite subsets in L ⊕ R , { A n } ∞ n =1 and { B n } ∞ n =2 .The desired properties of these sequences of sets are the following:1. A n is 2 n -separated 2 n -dense subset in C n [ s n − + 2 n , s n ]. If n = 1, this conditionis replaced by: A is a 2-separated 2-dense set in C .2. A n contains 2 n -separated 2 n -dense subsets in { ( B X i ( k ) (4 n ) , s n ) } nk =1 . . B n , n ≥ n -separated subset of C n ( s n − , s n − + 2 n ) such d ( A n , B n ) ≥ n , d ( A n − , B n ) ≥ n and A n − ∪ B n ∪ A n is a 2 n − -separated 2 n -dense subset in C n [ s n − + 2 n − , s n ]. If n = 2, the last condition is replaced by: 2-separated and2 -dense in C [ s , s ] = C .We construct such sets in steps. First we construct A n , then B n (for n ≥ A to be any 2-separated 2-dense subset of ( B X (4) , s ).The construction of A n ( n ≥
2) starts with picking a 2 n -separated 2 n -dense subsetof ( B X (4 n ) , s n ) . Then we gradually extend this subset to 2 n -separated 2 n -dense subsets of( B X i (2) (4 n ) , s n ) , . . . ( B X i ( n ) (4 n ) , s n ) . (8)Observe that our description of the sequence { i ( n ) } implies that many sets in thesequence (8) are the same.We complete the construction of A n extending the obtained set to a 2 n -separated2 n -dense subset of C n [ s n − + 2 n , s n ].To construct B n we remove from C n ( s n − , s n − +2 n ) all elements which are coveredby 2 n -balls centered in A n − ∪ A n . If the obtained set R is empty, we let B n = ∅ .Otherwise we let B n be a 2 n -separated 2 n -dense subset of R .The only condition which has to be verified is the condition that A n − ∪ B n ∪ A n is 2 n -dense in C n [ s n − + 2 n − , s n ] (and its version for n = 2). Here we use thecondition (7). By this condition, since A n − is 2 n − -dense in C n − [ s n − + 2 n − , s n − ],it is (cid:16) n − + (cid:0) (cid:1) n − (cid:17) -dense in C n [ s n − + 2 n − , s n − ]. Since (cid:16) n − + (cid:0) (cid:1) n − (cid:17) < n ,the conclusion follows from the construction of A n and B n .Let V = ∞ [ n =1 A n ! ∪ ∞ [ n =2 B n ! . We create a weighted graph with the vertex set V by joining a vertex v ∈ ( A n ∪ B n ), n ≥
2, to all vertices of n [ k =1 A k ! ∪ n [ k =2 B k ! which are within distance 3 · n to v in the normed space L ⊕ R . (Also we join eachvertex v ∈ A to all vertices of A which are within distance 6 to v .) The inequality(6) implies that in this way vertices of A n ∪ B n are joined only to some of the verticesin A n − ∪ B n − , A n ∪ B n , and A n +1 ∪ B n +1 . The vertex v is joined to some verticesin A n +1 ∪ B n +1 if v is within distance 3 · n +1 in L ⊕ R to those vertices. e assign weight (length) 2 n to all edges joining v ∈ ( A n ∪ B n ) with vertices of( A n − ∪ B n − ) S ( A n ∪ B n ). Thus we assign weight (length) 2 n +1 to the edges joining v ∈ ( A n ∪ B n ) with vertices of ( A n +1 ∪ B n +1 ). It is clear that we get a well-definedweighted graph. We denote the obtained weighted graph by W and endow it withits (weighted) shortest path metric.We estimate the number of edges incident to v ∈ ( A n ∪ B n ) in the following way.All vertices joined to v by edges are in a ball of radius 3 · n +1 centered at v . Thedistance between two vertices joined to v is at least 2 n − because all such verticesare in the set A n − ∪ B n − ∪ A n ∪ B n ∪ A n +1 ∪ B n +1 , and it is clear from the conditionson A n and B n that the set A n − ∪ B n − ∪ A n ∪ B n ∪ A n +1 ∪ B n +1 is 2 n − -separated.All of the elements of this set are in X i ( n +1) ⊕ R , and the dimension of this space is d ( n ) = i ( n + 1) + 1. Therefore X i ( n +1) ⊕ R -balls of radiuses 2 n − centered at pointsjoined to v with an edge have disjoint interiors and are contained in a ball of radius3 · n +1 + 2 n − centered at v . Comparing the volumes of the union of the 2 n − -balls and the (3 · n +1 + 2 n − )-ball containing them (it is the standard volumetricargument, see e.g. [MS86, Lemma 2.6]), we get that the number of vertices adjacentto v is at most (cid:18) · n +1 + 2 n − n − (cid:19) d ( n ) = 25 d ( n ) . Lemma 4.3.
The natural embedding of W into the normed space L ⊕ R has distor-tion ≤ . More precisely, its Lipschitz constant is ≤ , and the Lipschitz constantof the inverse map is ≤ .Proof. The statement about the Lipschitz constant of the natural embedding isimmediate. In fact, ends of an edge of length 2 n in W correspond to a vector in L ⊕ R whose length is ≤ · n .The fact that the Lipschitz constant of the inverse map is ≤ a and b be vertices of W . We may assume that b ∈ ( A n ∪ B n ) and a ∈ ( A k ∪ B k )for some k ≤ n . We use double induction. This means the following: First we provethe result for n = 1 using induction on ⌊|| a − b || / ⌋ (here we use Lemma 2.6).Next, we assume that the result holds for n = m and any k ≤ m . We show thatthis assumption can be used to prove the result for n = m + 1 using the inductionon ⌊|| a − b || / (2 m +1 ) ⌋ .So let us follow the described program. If we divide all distances by 2, the desiredinequality for a, b ∈ A is a special case of Lemma 2.6 for α = β = 1. Assumption:
Now we assume that we have proved the statement for all pairs b ∈ ( A m ∪ B m ), a ∈ ( A k ∪ B k ), k ≤ m .We show that this assumption can be used to prove the statement for b ∈ ( A m +1 ∪ B m +1 ), a ∈ ( A k ∪ B k ), k ≤ m + 1, using induction on ⌊|| a − b || / (2 m +1 ) ⌋ . f ⌊|| a − b || / (2 m +1 ) ⌋ ≤
2, then a and b are joined by an edge of length 2 m +1 . Inaddition, || a − b || ≥ m +1 (see the conditions 1 and 3 in the description of A n and B n ). So in this case we get the desired d W ( a, b ) ≤ || a − b || . Induction Hypothesis:
Suppose that the statement has been proved for all pairs b ∈ ( A m +1 ∪ B m +1 ), a ∈ ( A k ∪ B k ), k ≤ m + 1, and ⌊|| a − b || / (2 m +1 ) ⌋ ≤ D. (9)We show that this implies the same conclusion for pairs b ∈ ( A m +1 ∪ B m +1 ), a ∈ ( A k ∪ B k ), k ≤ m + 1, satisfying ⌊|| a − b || / (2 m +1 ) ⌋ ≤ D + 1.We need to consider the case where the inequality ( D + 2) · m +1 > || a − b || ≥ ( D + 1) · m +1 is satisfied, D ≥ D ∈ N . In such a case let e b be the vector on theline segment joining b to a at distance 2 · m +1 from b .It is clear from the construction that e b ∈ C m +1 . Therefore (see condition 3 in thelist of conditions on A n and B n ) there is a point b b ∈ V such that || b b − e b || ≤ m +1 .We have || b b − b || ≤ · m +1 and || b b − a || ≤ || b − a || − m +1 . (10)Also we have b b ∈ A m ∪ B m ∪ A m +1 ∪ B m +1 . Therefore d W ( b b, b ) ≤ m +1 .In the case when b b ∈ A m +1 ∪ B m +1 , we get the desired conclusion using the In-duction Hypothesis as follows: The inequality (10) implies that the pair a, b b satisfiesthe inequality (9). By the Induction Hypothesis, || a − b b || ≥ d W ( a, b b ) . Therefore || a − b || ≥ m +1 + || a − b b || ≥ m +1 + d W ( a, b b ) ≥ d W ( a, b ) . It remains to consider the case b b ∈ A m ∪ B m . This case is to be divided into twosubcases: k = m + 1 and k ≤ m . In the latter case we use the assumption that wehave proved the statement for points in ( S mi =1 A i ) ∪ ( S mi =2 B i ). In the former casewe use the Induction Hypothesis.The graph W contains ( B X k (2 n ) , k, n ∈ N , with all edges having the same weight of 2 n . This followsfrom the choice of 4 n as the diameter of the ball used in the construction of C n andthe fact that A n contains subsets which are 2 n -separated and 2 n -dense in the ball( B ( X k (4 n )) , s n ) for k ≤ i ( n ) (by combining these facts with our definitions). Weclaim that these ( B ( X k (2 n )) , W . This claim can be proved in the following way: Applying Lemma 2.6 we getthat the natural embeddings of the ( B ( X k (4 n )) , n )-graphs (constructed using the2 n -separated and 2 n -dense in the ball ( B ( X k (4 n )) , s n )) into L ⊕ R are uniformly ilipschitz. By Lemma 4.3, the natural embedding of W into L ⊕ R is also bilips-chitz. The conclusion on uniformity of bilipschitz embeddings of the ( B ( X k (2 n )) , W follows. Thus, by Lemma 2.7, bilipschitz embeddability of the graph W into a Banach space X implies that X admits uniformly isomorphic embeddingsof the spaces { X i } ∞ i =1 .Now we construct an unweighted graph H with maximum degree 3 whose exis-tence is claimed in Theorem 4.1. The graph H will admit a bilipschitz embeddinginto ( L ⊕ R ) ⊕ R . The graph H will be a modification of the weighted graph W . Asin the construction of Section 2, the graph H consists of long paths and short paths.Short paths correspond to vertices of W , long paths correspond to (weighted) edgesof W . The length of a short path corresponding to a vertex in A n ∪ B n is 2 · d ( n +1) .These lengths of short paths are chosen because they provide a sufficient number ofcolors for the coloring introduced in the next paragraph.We are going to color edges of W . For our purposes we need a proper edge coloring(that is, edges having a common end should have different colors). Of course, sincethe degrees of W are unbounded, we need infinitely many colors. Our purpose isto bound from above the number of colors used for edges incident with a vertex v ∈ A n ∪ B n by 2 · d ( n +1) . To achieve this goal we order vertices in V accordingto the magnitude of their R -coordinate in the decomposition L ⊕ R (starting withthose for which the R -coordinate is 0), resolving ties in an arbitrary way.We color edges incident with the first vertex arbitrarily (there are at most 25 d (1) ofsuch edges). For each of the further vertices in our list we need to color all uncolorededges incident with them. We do this according to the following procedure. Let v ∈ A n ∪ B n be the next vertex in our list. We pick an uncolored edge incident with v , let u be the other end of this edge. We cannot use for the edge vu the colorswhich have already been used for other edges incident to v and to u . There are atmost 25 d ( n ) − v . As for u , we know that(see the construction of W ) u ∈ A n − ∪ B n − ∪ A n ∪ B n ∪ A n +1 ∪ B n +1 , thereforethe degree of u is ≤ d ( n +1) . Therefore among 2 · d ( n +1) colors there should be anavailable color for the edge vu .Now we create a graph as in Section 2. The only difference is that we paste a longpath to the level corresponding to its color (rather than to the level correspondingto the path itself as we did in Section 2). The length of the long path correspondingto an edge of W of weight 2 n is M · n , where M is a positive integer which weare going to select now, together with the sequence { i ( n ) } ∞ n =1 (which, as we havealready mentioned, is a slowly increasing sequence). The main condition describingour choice of both objects is M · n > · d ( n +1) . (11)(Recall that d ( n ) = i ( n + 1) + 1.) This condition ensures that the length of a longpath is larger than the sum of the lengths of the short paths at its ends. e get an unweighted graph, let us denote it H . The maximum degree of H is 3because we use a proper edge coloring. The graph W admits a bilipschitz embeddinginto H : consider the map which maps each vertex of W to the vertex of level 1 onthe corresponding short path. The Lipschitz constant of this embedding is ≤ · M by (11). The Lipschitz constant of the inverse map is ≤ M − . In fact, if we considertwo vertices in the image of W , and join them by a shortest path in H , the pathgoes through some collection of short paths (possibly at the end vertices only). Thevertices in W corresponding to these short paths form a path in W . Each time theweight of the edge in W is M − × (the length of the corresponding path in H ). Theconclusion about the Lipschitz constant of the inverse map follows.Observe that the graph H is locally finite because its maximum degree is 3. Itremains to show that the graph H admits a bilipschitz embedding into any Banachspace X containing uniformly isomorphic { X i } . We do this by proving the factthat the graph H admits a bilipschitz embedding into ( L ⊕ R ) ⊕ R . By thefinite determination result of [Ost11c] this is enough because it is easy to see thatfinite dimensional subspaces of the space ( L ⊕ R ) ⊕ R are uniformly isomorphic tosubspaces in { X i } . Lemma 4.4.
The graph H admits a bilipschitz embedding into ( L ⊕ R ) ⊕ R .Proof. Vertices of W are in L ⊕ R . We map the short path corresponding to a vertex v ∈ A n ∪ B n onto those points of the line segment joining ( M v,
1) with (
M v, · d ( n +1) ) whose second coordinate is an integer. (The number M is introduced by(11). In a pair ( M v, a ) the first component is in L ⊕ R and the second componentis in the second R -summand.)Now we describe our map for long paths of H . For each long path of H , or, whatis the same, for each edge uv of W we pick a vector x uv ∈ L (the space L is identifiedwith the corresponding summand in ( L ⊕ R ) ⊕ R ). Suppose that the edge uv hasweight 2 n and color i in the coloring above. We number vertices of the long pathcorresponding to uv as u , u , . . . , u N , where N = M · n , u corresponds to u and v corresponds to u N . The image of u m ( m = 0 , , . . . , N ) under the map which weare constructing is given by T u m = ((cid:0)(cid:0) − mN (cid:1) M u + mN M v + mx uv , i (cid:1) if m ≤ N (cid:0)(cid:0) − mN (cid:1) M u + mN M v + ( N − m ) x uv , i (cid:1) if m ≥ N . (12)So we map vertices of the long path in H corresponding to an edge uv of W ontoa sequence of evenly distributed points in the union of two line segments joining( M u, i ) and (
M v, i ). We introduce also a map O given by Ou m = (cid:0) − mN (cid:1) M u + mN M v .The map T introduced by (12) is a Lipschitz map of the vertex set of H into( L ⊕ R ) ⊕ R for an arbitrary uniformly bounded set of vectors { x uv } in L . Toshow this it suffices to find a bound for the distances between images of ends ofan edge of H . For short-path-edges the distances are equal to 1 because their ends re mapped onto pairs of the form ( M v, i ), (
M v, i + 1). For a long-path-edge, thedistance between the ends is (cid:13)(cid:13)(cid:13)(cid:13) N M v − N M u ± x uv (cid:13)(cid:13)(cid:13)(cid:13) This norm can be estimated from above by 3 + sup uv || x uv || (we use the estimate forthe Lipschitz constant of Lemma 4.3).Therefore the purpose of a suitable selection of the set { x uv } is to ensure that T − is Lipschitz. In a similar situation in Section 2 we used Auerbach bases. Forthis construction we use a somewhat different type of biorthogonal sequences. Weuse systems whose existence was shown by Ovsepian-Pe lczy´nski [OP75]. We meanthe following result proved in [OP75] (see also [LT77, p. 44]): Theorem 4.5.
There is an absolute constant
C > such that for each separableBanach space Z , each sequence { f ∗ i } ⊂ Z ∗ , and each sequence { f i } ⊂ Z there existsa biorthogonal sequence { z i , z ∗ i } ∞ i =1 in Z such that || z i || ≤ C , || z ∗ i || ≤ C , the linearspan of { z i } contains the sequence { f i } , and the linear span of { z ∗ i } contains thesequence { f ∗ i } .Remark . Pe lczy´nski [Pel76] and Plichko [Pli76] proved that the constant C canbe chosen to be an arbitrary number > Z = L in the following situation. We form the sequence { f ∗ i } in the following way. We denote by P L : L ⊕ R → L the natural projection.For any two edges uv and wz in W we consider all vectors of the form P L ( Ou m − Ow p ) (13)for all admissible values of m and p . The map O was defined after formula (12) and Ow p is the image of a vertex w p of a long path corresponding to wz .For each vector of the form (13) we pick a supporting functional f ∗ ∈ L ∗ , thatis, a functional f ∗ satisfying f ∗ ( P L ( Ou m − Ow p )) = || P L ( Ou m − Ow p ) || and || f ∗ || = 1. There are countably many such functionals, so we can form asequence { f ∗ i } containing all of them. Also we form a sequence { f i } containing allof the vectors of the form (13).Now we describe a suitable choice of the vectors x uv for (12). We enumerateedges of W in the non-decreasing order of the larger R -coordinates of their ends,resolving ties arbitrarily. Let uv be the first edge in the ordering. We pick as x uv an element of the sequence { z i } satisfying the following two conditions: • z i is annihilated by all functionals f ∗ j of the sequence { f ∗ i } supporting vectors P L ( Ou m − Ow p ), where u m is in the long path corresponding to uv and w p is inthe long path corresponding to an edge wz for which the smaller R -coordinateof its ends is ≤ s . x ∗ uv := z ∗ i annihilates all vectors f j of the sequence { f i } of the form P L ( Ou m − Ow p ), where u m is in the long path corresponding to uv and w p is in the longpath corresponding to an edge wz for which the smaller R -coordinate of itsends is ≤ s .Such pair z i , z ∗ i exists because there are finitely many f j and f ∗ j satisfying theconditions. (Here we use the following conditions of Theorem 4.5: the linear span of { z i } contains the sequence { f i } , and the linear span of { z ∗ i } contains the sequence { f ∗ i } .)We make a similar choice of x uv for all further edges in the ordering. More details:If we consider an edge for which the larger R -coordinate of an end is in the interval( s n − , s n ], we replace s by s n +2 in the conditions above. Also we pick different pairs z i , z ∗ i for different edges uv .With this choice of vectors x uv , let us estimate from above the Lipschitz constantof the inverse of T . Let x, y be two vertices of H . Let x be on a long path joining u and v and y be on a long path joining w and z (this is a generic description because x and y are allowed to be the end vertices of the long paths). We need to estimatefrom above the quotient d H ( x, y ) || T x − T y || . (14)We consider a shortest xy -path in H . It has naturally defined short-path portion and long-path portion . There are two cases: (1) The length of the short-path portionof this path has length ≥ d H ( x, y ); The length of the long-path portion of of thispath has length > d H ( x, y ).The construction of the graph H (see inequality (11)) is such that in the case (1)the short-path portion consists of just one piece. Let the short path portion startat level (color) i and end at level j . Then | i − j | ≥ d H ( x, y ). On the other hand,since the sum ( L ⊕ R ) ⊕ R is direct, we have || T x − T y || ≥ | i − j | ≥ d H ( x, y ).In the case (2) we ignore the difference in the second R -coordinate (caused bythe different colors of the edges uv and wz ). There are two subcases: Subcase (A):The vertices x and y are on the same long path; Subcase (B): The vertices x and y are on different long paths. Subcase (A):
Observe that our construction is such that there is a functionalsupporting P L ( u − v ) (let us denote it f ∗ uv ) which annihilates by x uv (because u − v is of the form Ou m − Ow p ). Let x = u p , y = u s , we have d H ( x, y ) = | s − p | and || T u p − T u s || ≥ | f ∗ uv P L ( Ou p − Ou s ) | = M | s − p | N || P L ( u − v ) || . If || P L ( u − v ) || = || P L u − P L v || is a nontrivially large part of || u − v || ≥ NM , weget the desired estimate. f || P L u − P L v || is only a small part of || u − v || , then the difference between the R -coordinates of u and v is the large part of || u − v || . Denoting the projection of L ⊕ R to R by P R , we use P R ( x uv ) = 0 and get || T u p − T u s || ≥ M | s − p | N | P R u − P R v | , so we get the estimate in this case, too. Subcase (B):
Let x be on a long path corresponding to an edge uv in W , and y be on a long path corresponding to wz . Then there are two possibilities:(i) One of the edges uv and wz has the largest R -coordinate of its ends in theinterval ( s n − , s n ], and the other edge has the least R -coordinate of its ends in theinterval [ s m , s m +1 ), where m ≥ n + 2(ii) It is not the case. Subsubcase (i):
Let x = u m and y = w p . Ignoring the second R -coordinate (inthe sum L ⊕ R ⊕ R ) and using P R x uv = P R x wz = 0 we get || T x − T y || ≥ M | P R u − P R w | ≥ M ( s m − s n ) , (15)where u and w are the corresponding vertices picked in such a way that u has larger R -coordinate than v and w has smaller R -coordinate than w . On the other hand d H ( x, y ) ≤ d H ( u, w ) + M n + M m +1 ≤ M || u − w || + M n + M m +1 ≤ M | P R u − P R w | + M n + M m +1 + M n + M m +1 ≤ M | P R u − P R w | + M ( s m − s n ) ≤ M | P R u − P R w | . To get these inequalities we use • The triangle inequality in H for the first inequality. • Lemma 4.3 for the second inequality. • The observation that || P L z || ≤ n if z ∈ C n for the third inequality (see thedefinition of C n ). • The gap condition (6) for the fourth inequality. • The second inequality in (15) for the fifth inequality.The conclusion follows.
Subsubcase (ii):
We may assume without loss of generality that x is closer (in H )to the short path corresponding to u than to the short path corresponding to v and hat y is closer to the short path corresponding to w rather than to the short pathcorresponding to z . We have || T x − T y ||≥ (cid:13)(cid:13)(cid:13)(cid:16) − mN (cid:17) M u + mN M v + mx uv − (cid:16) − pN ′ (cid:17) M w − pN ′ M z − px wz (cid:13)(cid:13)(cid:13) , (16)where N ′ is the length of the long path corresponding to wz . Let us denote thevector whose norm is taken in the right-hand side of (16) by B . We get || T x − T y || ≥ | x ∗ uv ( B ) | = m. || T x − T y || ≥ | x ∗ wz ( B ) | = p. Writing x ∗ uv ( B ) , x ∗ wz ( B ) we mean that the functionals x ∗ uv , x ∗ wz ∈ L ∗ act on a vector t ∈ L ⊕ R by acting on P L t . We use the fact that the difference (cid:0) − mN (cid:1) M u + mN M v − (cid:0) − pN ′ (cid:1) M w − pN ′ M z is of the from Ou m − Ow p for edges satisfying theconditions above. Thus this difference is annihilated by x ∗ uv and x ∗ wz .We apply the triangle inequality to (16) and get || T x − T y ||≥ M || u − w || − mMN || u − v || − pMN ′ || w − z || − mC − pC = M || u − w || − m (cid:18) MN || u − v || + C (cid:19) − p (cid:18) MN ′ || w − z || + C (cid:19) ≥ d H ( u, w ) − m (cid:18) MN || u − v || + C (cid:19) − p (cid:18) MN ′ || w − z || + C (cid:19) , where C = sup i || z i || = sup u,v || x uv || . (We used Lemma 4.3 to get the last inequality.)Observe that the numbers in brackets in the last line are bounded by an absoluteconstant, let us denote it by D . We also have d H ( x, y ) ≤ d H ( u, w )+ m + p . Therefore d H ( x, y ) || T x − T y || ≤ min (cid:26) d H ( u, w ) + m + pd H ( u, w ) − D ( m + p ) , d H ( u, w ) + m + pm , d H ( u, w ) + m + pp (cid:27) . It is easy to see that the minimum in the last formula is bounded from above by anabsolute constant. [Aue30] H. Auerbach,
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