Markov type of Alexandrov spaces of nonnegative curvature
aa r X i v : . [ m a t h . M G ] M a y Markov type of Alexandrov spacesof nonnegative curvature ∗† Shin-ichi OHTA ‡ Department of Mathematics, Faculty of Science,Kyoto University, Kyoto 606-8502, JAPAN e-mail : [email protected]
Abstract
We prove that Alexandrov spaces of nonnegative curvature have Markov type2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subsetof an Alexandrov space of nonnegative curvature into a 2-uniformly convex Banachspace is extended to a Lipschitz continuous map on the entire space.
The aim of the present article is to contribute to the nonlinearization of the geometry ofBanach spaces from the viewpoint of metric geometry. Among them, our main object isMarkov type of metric spaces due to Ball.Markov type is a generalization of Rademacher type of Banach spaces. Rademachertype and cotype describe the behaviour of sums of independent random variables in Ba-nach spaces, and these properties have fruitful analytic and geometric applications (cf.[LT] and [MS]). Enflo [E] first gave a generalized notion of type of metric spaces, which iscalled Enflo type now, and a variant of Enflo type was studied by Bourgain, Milman andWolfson [BMW]. After them, Ball [B] introduced the notion of Markov type of metricspaces, and showed its importance in connection with the extension problem of Lipschitzmaps. He showed that any Lipschitz continuous map from a subset of metric space X having Markov type 2 into a reflexive Banach space having Markov cotype 2 can be ex-tended to a Lipschitz map on the entire space X . Here Markov cotype of Banach spacesis a notion also introduced by Ball. It is worthwhile to mention that how to formulatea notion of cotype for general metric spaces has been an important question, we referto [MN3] for a recent breakthrough on this topic. Markov type has found further deepapplications in the extension problem of Lipschitz maps ([NPSS], [MN1]) as well as in ∗ Mathematics Subject Classification (2000): 46B20, 53C21, 60J10. † Keywords: Markov type, Lipschitz map, Alexandrov space. ‡ Partly supported by the JSPS fellowship for research abroad. under the reverse curvature bound .Our main theorem asserts that Alexandrov spaces of nonnegative curvature haveMarkov type 2 with a universal estimate on the Markov type constant (Theorem 4.2).This theorem gives us first and rich examples of positively curved spaces having Markovtype 2. As an immediate corollary by virtue of Ball’s extension theorem, any Lipschitzcontinuous map from a subset of an Alexandrov space of nonnegative curvature into areflexive Banach space having Markov cotype 2 can be extended to a Lipschitz continuousmap on the entire space (Corollary 4.5). In particular, our estimate on the ratio of theLipschitz constants is independent of the dimension. Compare this with [LS], [LPS] and[LN]. Our key tool is the inequality (3.1) in Theorem 3.3 due to Sturm.The article is organized as follows. We briefly review the theories of linear and nonlin-ear types and Alexandrov spaces of nonnegative curvature in Sections 2 and 3, respectively.Section 4 is devoted to the proof of the main theorem. Finally, in Section 5, we give ashort remark on nonlinearizations of the 2-uniform smoothness and convexity of Banachspaces in connection with curvature bounds in metric geometry.
Acknowledgements.
I would like to express my gratitude to Assaf Naor and Yuval Peresfor their valuable comments on the first version of the paper. Their suggestions exceed-ingly improved the presentation of the paper (see Remark 4.3 and Proposition 5.3). Thiswork was completed while I was visiting Institut f¨ur Angewandte Mathematik, Universit¨atBonn. I am grateful to the institute for its hospitality.
In this section, we recall Rademacher type and cotype of Banach spaces and severalextensions of Rademacher type to nonlinear spaces. We refer to [LT] and [MS] for basicfacts on Rademacher type and cotype. Throughout the article, we restrict ourselves tothe case of p = 2, i.e., we will treat only type 2 and cotype 2.A Banach space ( V, k· k ) is said to have Rademacher type K ≥ N ∈ N and { v i } Ni =1 ⊂ V , we have12 N X ε ∈{− , } N (cid:13)(cid:13)(cid:13)(cid:13) N X i =1 ε i v i (cid:13)(cid:13)(cid:13)(cid:13) ≤ K N X i =1 k v i k , (2.1)where ε = ( ε i ) Ni =1 . A fundamental example of a space possessing Rademacher type 2 isa 2-uniformly smooth Banach space. A Banach space ( V, k · k ) is said to be 2 -uniformlysmooth (or, equivalently, have modulus of smoothness of power type
2) if there is a constant2 ≥ (cid:13)(cid:13)(cid:13)(cid:13) v + w (cid:13)(cid:13)(cid:13)(cid:13) ≥ k v k + 12 k w k − S k v − w k (2.2)holds for all v, w ∈ V (see [BCL]). The infimum of such a constant S is denoted by S ( V ).For instance, for 2 ≤ p < ∞ , an L p -space L p ( Z ) over an arbitrary measure space Z is2-uniformly smooth with S = √ p −
1, and hence it has Rademacher type 2. Note that,if V is a Hilbert space, then the parallelogram identity yields equality in (2.2) with S = 1.Rademacher cotype 2 and the 2-uniform convexity of a Banach space are definedsimilarly by replacing (2.1) and (2.2) with12 N X ε ∈{− , } N (cid:13)(cid:13)(cid:13)(cid:13) N X i =1 ε i v i (cid:13)(cid:13)(cid:13)(cid:13) ≥ K N X i =1 k v i k , (2.3) (cid:13)(cid:13)(cid:13)(cid:13) v + w (cid:13)(cid:13)(cid:13)(cid:13) ≤ k v k + 12 k w k − C k v − w k , (2.4)respectively. Denote by C ( V ) the least constant C ≥ < p ≤ L p ( Z ) is2-uniformly convex with C = 1 / √ p − L ( Z ) has Rademacher cotype 2, though it is not 2-uniformly convex.The first nonlinear extension of Rademacher type was given by Enflo. Definition 2.1 (Enflo type, [E]) A metric space (
X, d ) is said to have
Enflo type K ≥ N ∈ N and { x ε } ε ∈{− , } N ⊂ X , it holds that X ε ∈{− , } N d ( x ε , x − ε ) ≤ K X ε ∼ ε ′ d ( x ε , x ε ′ ) , (2.5)where ε = ( ε i ) Ni =1 and ε ∼ ε ′ holds if P Ni =1 | ε i − ε ′ i | = 2 (i.e., ε and ε ′ are adjacent). Theleast such a constant K ≥ E ( X ).By taking x ε = P Ni =1 ε i v i , we easily see that Enflo type 2 implies Rademacher type 2for Banach spaces. However, the converse is not known in general. See [NS] for a partialpositive result and [MN2] for related work.We next recall Markov type introduced by Ball. As is indicated in its name, we usea Markov chain to define Markov type. For N ∈ N , consider a stationary, reversibleMarkov chain { M l } l ∈ N ∪{ } on the state space { , , . . . , N } with transition probabilities a ij := Pr( M l +1 = j | M l = i ). Namely, if we set π i := Pr( M = i ), then { π i } Ni =1 and A = ( a ij ) Ni,j =1 satisfy0 ≤ π i ≤ , ≤ a ij ≤ , N X i =1 π i = 1 , N X j =1 a ij = 1 , π i a ij = π j a ji (2.6)for all i, j = 1 , , . . . , N . The third and fourth inequalities guarantee the stationariness( P Ni =1 π i a ij = π j ) and the reversibility of the Markov chain { M l } l ∈ N ∪{ } .3 efinition 2.2 (Markov type, [B, Definition 1.3]) A metric space ( X, d ) is said to have
Markov type K ≥ α ∈ (0 , N ∈ N , { x i } Ni =1 ⊂ X , { π i } Ni =1 and A = ( a ij ) Ni,j =1 satisfying (2.6), we have(1 − α ) N X i,j =1 π i c ij d ( x i , x j ) ≤ K α N X i,j =1 π i a ij d ( x i , x j ) , (2.7)where we set C = ( c ij ) Ni,j =1 = (1 − α )( I − αA ) − and I stands for the identity matrix.The infimum of K ≥ M ( X ).We remark that Ball’s original definition concerns only the case of π i ≡ N − . Theabove slightly extended (but equivalent) formulation can be found in [NPSS]. Note that C = (1 − α )( I − αA ) − = (1 − α ) ∞ X l =0 α l A l . Hence C = ( c ij ) Ni,j =1 also satisfies0 ≤ c ij ≤ , N X j =1 c ij = 1 , π i c ij = π j c ji for all i, j = 1 , , . . . , N .We recall some important properties of Markov type. Markov type has an equivalentform which is more convenient in some circumstances. For l ∈ N and A = ( a ij ) Ni,j =1 , weset A l = ( a ( l ) ij ) Ni,j =1 . In particular, a (1) ij = a ij . Theorem 2.3 ([B, Theorem 1.6])
Let ( X, d ) be a metric space and assume that there isa constant K ≥ such that the inequality N X i,j =1 π i a ( l ) ij d ( x i , x j ) ≤ K l N X i,j =1 π i a ij d ( x i , x j ) (2.8) holds for all l ∈ N , N ∈ N , { x i } Ni =1 ⊂ X , { π i } Ni =1 and A = ( a ij ) Ni,j =1 satisfying ( ) .Then ( X, d ) has Markov type with M ( X ) ≤ K . Conversely, if ( X, d ) has Markov type , then ( X, d ) satisfies ( ) with K = 2 √ eM ( X ) . Markov type is known to be strong enough for implying Enflo type.
Proposition 2.4 ([NS, Proposition 1])
If a metric space ( X, d ) has Markov type , thenit has Enflo type . To state Ball’s theorem which guarantees the usefulness of Markov type, we need todefine Markov cotype of Banach spaces also introduced by Ball.4 efinition 2.5 (Markov cotype, [B, Definition 1.5]) A Banach space ( V, k · k ) is said tohave Markov cotype K ≥ α ∈ (0 , N ∈ N , { v i } Ni =1 ⊂ V and A = ( a ij ) Ni,j =1 satisfying (2.6) with π i ≡ N − , we have α N X i,j =1 a ij (cid:13)(cid:13)(cid:13)(cid:13) N X k =1 ( c ik − c jk ) v k (cid:13)(cid:13)(cid:13)(cid:13) ≤ K (1 − α ) N X i,j =1 c ij k v i − v j k , where we set C = ( c ij ) Ni,j =1 = (1 − α )( I − αA ) − . We denote by N ( V ) the infimum ofsuch a constant K ≥ L ( Z )has Rademacher cotype 2 and does not have Markov cotype 2 (see [B]). It is known thata 2-uniformly convex Banach space ( V, k · k ) has Markov cotype 2 with N ( V ) ≤ C ( V )([B, Theorem 4.1]). For a Lipschitz continuous map f : X −→ Y between metric spaces,we denote by Lip ( f ) its Lipschitz constant, that is, Lip ( f ) := sup x,y ∈ X, x = y d Y ( f ( x ) , f ( y )) d X ( x, y ) . Theorem 2.6 ([B, Theorem 1.7])
Let ( X, d ) be a metric space having Markov type and ( V, k · k ) be a reflexive Banach space having Markov cotype . Then, for any Lipschitzcontinuous map f : Z −→ V from a subset Z ⊂ X , there exists a Lipschitz continuousextension ˜ f : X −→ V of f with Lip ( ˜ f ) ≤ M ( X ) N ( V ) Lip ( f ) . In particular, if ( V, k · k ) is a -uniformly convex Banach space, then we have Lip ( ˜ f ) ≤ M ( X ) C ( V ) Lip ( f ) . We refer to [BLMN], [LMN], [MN1] and [NPSS] for further applications of Markovtype. We end this section with several examples of spaces having Markov type.
Example 2.7 (i) (Hilbert spaces, [B, Proposition 1.4]) A Hilbert space ( H, h· , ·i ) hasMarkov type 2 with M ( H ) = 1.(ii) (Products) For two metric spaces ( X , d ) and ( X , d ) having Markov type 2, let( X, d ) be the l -product of them, that is, X := X × X and d (cid:0) ( x , x ) , ( y , y ) (cid:1) := { d ( x , y ) + d ( x , y ) } / for ( x , x ) , ( y , y ) ∈ X . Then ( X, d ) has Markov type 2 with M ( X ) ≤ max { M ( X ) , M ( X ) } . (iii) (The bi-Lipschitz equivalence) Given two metric spaces ( X , d ) and ( X , d ), if( X , d ) has Markov type 2 and if there is a bi-Lipschitz homeomorphism f : X −→ X ,then ( X , d ) has Markov type 2 with M ( X ) ≤ Lip ( f ) Lip ( f − ) M ( X ) . { ( X i , d i ) } ∞ i =1 converges to a (pointed) metric space ( X, d ) in the sense of the (pointed) Gromov-Hausdorff convergence and if every ( X i , d i ) has Markov type 2 with lim inf i →∞ M ( X i ) < ∞ , then ( X, d ) has Markov type 2 with M ( X ) ≤ lim inf i →∞ M ( X i ) . (v) (2-uniformly smooth Banach spaces, [NPSS, Theorem 1.2]) A 2-uniformly smoothBanach space ( V, k · k ) has Markov type 2 with M ( V ) ≤ S ( V ).(vi) (Trees and hyperbolic groups, [NPSS, Theorem 1.4, Corollary 1.6]) There existsa universal constant C t for which every tree T with arbitrary positive edge lengths hasMarkov type 2 with M ( T ) ≤ C t . There also exists a universal constant C h such thatevery δ -hyperbolic group has Markov type 2 with M ≤ C h (1 + δ ). More precisely, we fix apresentation of the group and consider its Cayley graph G equipped with the word metric.If G is δ -hyperbolic as a metric space, then it has Markov type 2 with M ( G ) ≤ C h (1 + δ ).Naor et al. have obtained an estimate for general δ -hyperbolic metric spaces, and it impliesthe above results.(vii) (Riemannian manifolds with pinched negative sectional curvature, [NPSS, The-orem 1.7]) An n -dimensional, complete and simply connected Riemannian manifold M has Markov type 2 if its sectional curvature takes values in [ − R, − r ] for some R > r > M ( M ) is estimated from above by using n , r and R .(viii) (Laakso graphs, [NPSS, Proposition 7.1]) The Laakso graphs ([La]) have Markovtype 2. In this section, we recall the definition of Alexandrov spaces of nonnegative curvature.We refer to [BGP] and [BBI] as standard references.A metric space (
X, d ) is said to be geodesic if every two points x, y ∈ X can beconnected by a curve γ : [0 , −→ X from x to y with length( γ ) = d ( x, y ). A rectifiablecurve γ : [0 , −→ X is called a geodesic if it is locally minimizing and has a constantspeed. A geodesic γ : [0 , −→ X is said to be minimal if it satisfies length( γ ) = d ( γ (0) , γ (1)). Definition 3.1
A geodesic metric space (
X, d ) is called an
Alexandrov space of nonnega-tive curvature if, for all three points x, y, z ∈ X and any minimal geodesic γ : [0 , −→ X between y and z , we have d (cid:18) x, γ (cid:18) (cid:19)(cid:19) ≥ d ( x, y ) + 12 d ( x, z ) − d ( y, z ) . We recall some examples. Every example gives a new class of metric spaces havingMarkov type 2. 6 xample 3.2 (i) A complete Riemannian manifold is an Alexandrov space of nonneg-ative curvature if and only if its sectional curvature is nonnegative everywhere. In par-ticular, spheres, tori and symmetric spaces of compact type are Alexandrov spaces ofnonnegative curvaure.(ii) For a compact convex domain Ω ⊂ R n , let X = ∂ Ω equip the length metric d in-duced from the standard metric of R n . Then ( X, d ) is an Alexandrov space of nonnegativecurvature.(iii) Let (
M, g ) be a Riemannian manifold of nonnegative sectional curvature and G be a compact group acting on M by isometries. Then the quotient space M/G equippedwith the quotient metric is an Alexandrov space of nonnegative curvature.There is a rich and deep theory on the geometry and analysis on Alexandrov spaces,but all we need in the present paper is the following characterization due to Sturm (seealso [LS, Proposition 3.2]).
Theorem 3.3 ([S, Theorem 1.4])
A geodesic metric space ( X, d ) is an Alexandrov spaceof nonnegative curvature if and only if, for any N ∈ N , { x i } Ni =1 ⊂ X , y ∈ X and { a i } Ni =1 ⊂ [0 , with P Ni =1 a i = 1 , we have N X i,j =1 a i a j { d ( x i , x j ) − d ( x i , y ) − d ( x j , y ) } ≤ . (3.1)More probabilistically speaking, the inequality (3.1) says that, for any finitely sup-ported X -valued random variable Z and its independent copy e Z , E [ d ( Z, e Z ) ] ≤ E [ d ( Z, y ) ]holds for all y ∈ X .We also remark that the inequality (3.1) corresponds to the following fact in a Hilbertspace ( H, h· , ·i ). For any N ∈ N , { v i } Ni =1 ⊂ H and { a i } Ni =1 ⊂ [0 ,
1] with P Ni =1 a i = 1, N X i,j =1 a i a j {k v i − v j k − k v i − w k − k v j − w k } = 2 N X i,j =1 a i a j h v i − w, w − v j i = 2 (cid:28)(cid:18) N X i =1 a i v i (cid:19) − w, w − (cid:18) N X j =1 a j v j (cid:19)(cid:29) = − (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) N X i =1 a i v i (cid:19) − w (cid:13)(cid:13)(cid:13)(cid:13) ≤ w ∈ H . In this section, we prove our main theorem. Throughout the section, let (
X, d ) be anAlexandrov space of nonnegative curvature, and fix N ∈ N , { x i } Ni =1 ⊂ X , { π i } Ni =1 and7 = ( a ij ) Ni,j =1 satisfying (2.6). For 1 ≤ i, j ≤ N and l ∈ N , set d ij := d ( x i , x j ) and E ( l ) := N X i,j =1 π i a ( l ) ij d ij . Recall the notation A l = ( a ( l ) ij ) Ni,j =1 and that (2.6) implies0 ≤ a ( l ) ij ≤ , N X j =1 a ( l ) ij = 1 , π i a ( l ) ij = π j a ( l ) ji for all i, j = 1 , , . . . , N . Lemma 4.1
For any l ∈ N , we have E (2 l ) ≤ E ( l ) .Proof. We calculate E (2 l ) = N X i,j,k =1 π i a ( l ) ik a ( l ) kj d ij = N X i,j,k =1 π k a ( l ) ki a ( l ) kj ( d ki + d kj + d ij − d ki − d kj )= N X i,j,k =1 π k a ( l ) ki a ( l ) kj ( d ki + d kj ) + N X k =1 π k (cid:26) N X i,j =1 a ( l ) ki a ( l ) kj ( d ij − d ki − d kj ) (cid:27) . Since P Ni =1 a ( l ) ki = 1, we have N X i,j,k =1 π k a ( l ) ki a ( l ) kj ( d ki + d kj ) = N X i,k =1 π k a ( l ) ki d ki + N X j,k =1 π k a ( l ) kj d kj = 2 E ( l ) . Moreover, applying Theorem 3.3 with a i = a ( l ) ki and y = x k , we obtain N X i,j =1 a ( l ) ki a ( l ) kj ( d ij − d ki − d kj ) ≤ k . This completes the proof. ✷ Theorem 4.2
Let ( X, d ) be an Alexandrov space of nonnegative curvature. Then ( X, d ) has Markov type with M ( X ) ≤ √ . More precisely, ( X, d ) satisfies the inequality ( ) with K = 1 + √ .Proof. We will prove the theorem by induction. Note that the triangle inequality implies p E ( l + m ) ≤ p E ( l )+ p E ( m ) for all l, m ∈ N , and that Lemma 4.1 yields E (2 n ) ≤ n E (1)8or all n ∈ N . Assume that E ( l ) ≤ (1 + √ l E (1) holds for all 1 ≤ l ≤ n for fixed n ∈ N .Then, for 2 n + 1 ≤ l ≤ n +1 , take t ∈ (0 ,
1] with l = (1 + t )2 n and observe p E ( l ) ≤ p E (2 n ) + p E ( t n ) ≤ p n E (1) + (1 + √ p t n E (1)= { √ √ t } p n E (1) ≤ (1 + √ √ t p n E (1)= (1 + √ p l E (1) . Here the fourth implication follows from the fact that the function f ( t ) = (1 + √ √ t − √ t )is decreasing in t ∈ (0 ,
1] and f (1) = 1. ✷ Remark 4.3
The author’s original proof used Lemma 4.1 as well as the inequality α l E (2 l ) + 2 α l +1 E (2 l + 1) + α l +2 E (2 l + 2) ≤ α ) α l { α l E ( l ) + α l +1 E ( l + 1) } for l ∈ N and α ∈ (0 , M ( X ) ≤ √
6. After thefirst version of this paper was completed, the author learned the above simpler, improvedproof from A. Naor and Y. Peres.We have two corollaries by virtue of Proposition 2.4 and Theorem 2.6.
Corollary 4.4
Let ( X, d ) be an Alexandrov space of nonnegative curvature. Then ( X, d ) has Enflo type . Corollary 4.5
Let ( X, d ) be an Alexandrov space of nonnegative curvature and ( V, k · k ) be a reflexive Banach space having Markov cotype . Then, for any Lipschitz continuousmap f : Z −→ V from a subset Z ⊂ X , there exists a Lipschitz continuous extension ˜ f : X −→ V of f with Lip ( ˜ f ) ≤ √ N ( V ) Lip ( f ) . In particular, if ( V, k · k ) is -uniformly convex, then we have Lip ( ˜ f ) ≤ √ C ( V ) Lip ( f ) . We mention that our bound of the ratio of Lipschitz constants is independent of thedimension of X . Compare this with [LN, Theorem 1.6]. This section is devoted to a short remark toward a nonlinearization of the 2-uniformsmoothness (and convexity). As we have already seen in (2.2), the 2-uniform smoothnessof a Banach space is defined by using the inequality (cid:13)(cid:13)(cid:13)(cid:13) v + w (cid:13)(cid:13)(cid:13)(cid:13) ≥ k v k + 12 k w k − S k v − w k . (5.1)9y replacing v and w with w + v and w − v , this inequality is rewritten as (cid:13)(cid:13)(cid:13)(cid:13) v + w (cid:13)(cid:13)(cid:13)(cid:13) ≤ S k v k + 12 k w k − k v − w k . (5.2)Natural generalizations of (5.1) and (5.2) would be the following: Let ( X, d ) be ageodesic metric space. For any three points x, y, z ∈ X and minimal geodesic γ : [0 , −→ X from y to z , we have d (cid:18) x, γ (cid:18) (cid:19)(cid:19) ≥ d ( x, y ) + 12 d ( x, z ) − S d ( y, z ) (5.3)or d (cid:18) x, γ (cid:18) (cid:19)(cid:19) ≤ S d ( x, y ) + 12 d ( x, z ) − d ( y, z ) . (5.4)We will say that a geodesic metric space ( X, d ) satisfies ( ) (or (5.4)) if (5.3) (or (5.4))holds for all x, y, z ∈ X and all minimal geodesic γ : [0 , −→ X from y to z . On onehand, the inequality (5.3) generalizes the nonnegatively curved property in the sense ofAlexandrov which corresponds to the case of S = 1 (see Section 3). On the other hand,the inequality (5.4) extends the CAT(0)-inequality which amounts to the case of S = 1(cf. [BH]). This is a reason why both negatively and positively curved spaces have Markovtype 2. Compare Example 2.7 and Theorem 4.2.We mention that we can also regard (5.1) as an upper curvature bound of the unitsphere (see [O1]), and that the reverse inequality of (5.3) (a generalized 2-uniform con-vexity) has been studied in [O2].As an application of the inequality (5.4), we give an example of a nonlinear and non-Riemannian (in other words, Finslerian) space possessing Enflo type 2. We first prove alemma. Lemma 5.1
Let a geodesic metric space ( X, d ) satisfy ( ) . Then, for any four points w, x, y, z ∈ X , we have d ( w, y ) + d ( x, z ) ≤ S { d ( w, x ) + d ( y, z ) } + d ( w, z ) + d ( y, x ) . (5.5) Proof.
Take a minimal geodesic γ : [0 , −→ X between x and z . Then (5.4) yields that d (cid:18) w, γ (cid:18) (cid:19)(cid:19) ≤ S d ( w, x ) + 12 d ( w, z ) − d ( x, z ) ,d (cid:18) y, γ (cid:18) (cid:19)(cid:19) ≤ S d ( y, z ) + 12 d ( y, x ) − d ( x, z ) . Thus we see d ( w, y ) ≤ (cid:26) d (cid:18) w, γ (cid:18) (cid:19)(cid:19) + d (cid:18) γ (cid:18) (cid:19) , y (cid:19)(cid:27) ≤ (cid:26) d (cid:18) w, γ (cid:18) (cid:19)(cid:19) + d (cid:18) γ (cid:18) (cid:19) , y (cid:19) (cid:27) ≤ S { d ( w, x ) + d ( y, z ) } + d ( w, z ) + d ( y, x ) − d ( x, z ) . This is the required inequality. ✷ roposition 5.2 If a geodesic metric space ( X, d ) satisfies ( ) , then it has Enflo type with E ( X ) ≤ S . In particular, a CAT(0) -space ( X, d ) has Enflo type with E ( X ) = 1 ,and a -uniformly smooth Banach space ( V, k · k ) has Enflo type with E ( V ) ≤ S ( V ) .Proof. We shall prove by induction in N ∈ N . In the case of N = 1, for any { x , x − } ⊂ X , we immediately see d ( x , x − ) + d ( x − , x ) ≤ S { d ( x , x − ) + d ( x − , x ) } . Fix N ≥ { x δ } δ ∈{− , } N − ⊂ X , it holds that X δ ∈{− , } N − d ( x δ , x − δ ) ≤ S X δ ∼ δ ′ d ( x δ , x δ ′ ) , where δ = ( δ i ) N − i =1 and δ ∼ δ ′ holds if P N − i =1 | δ i − δ ′ i | = 2. Now we choose an arbitrary { x ε } ε ∈{− , } N ⊂ X . For each δ ∈ {− , } N − , Lemma 5.1 implies d ( x ( δ, , x ( − δ, − ) + d ( x ( δ, − , x ( − δ, ) ≤ S { d ( x ( δ, , x ( δ, − ) + d ( x ( − δ, − , x ( − δ, ) } + d ( x ( δ, , x ( − δ, ) + d ( x ( − δ, − , x ( δ, − ) . Summing up this inequality in δ ∈ {− , } N − , we have X ε ∈{− , } N d ( x ε , x − ε ) ≤ S X δ ∈{− , } N − { d ( x ( δ, , x ( δ, − ) + d ( x ( δ, − , x ( δ, ) } + X δ ∈{− , } N − { d ( x ( δ, , x ( − δ, ) + d ( x ( δ, − , x ( − δ, − ) } . By our assumption, the second term in the right-hand side is estimated as X δ ∈{− , } N − { d ( x ( δ, , x ( − δ, ) + d ( x ( δ, − , x ( − δ, − ) }≤ S X δ ∼ δ ′ { d ( x ( δ, , x ( δ ′ , ) + d ( x ( δ, − , x ( δ ′ , − ) } . Therefore we obtain X ε ∈{− , } N d ( x ε , x − ε ) ≤ S X ε ∼ ε ′ d ( x ε , x ε ′ ) . This completes the proof. ✷ The following observation (in connection with [FS]) is due to A. Naor.
Proposition 5.3
Let ( X, d ) be a metric space satisfying the Ptolemy inequality, that is,for all four points w, x, y, z ∈ X , we have d ( w, y ) · d ( x, z ) ≤ d ( w, x ) · d ( y, z ) + d ( w, z ) · d ( y, x ) . (5.6) Then ( X, d ) satisfies the inequality ( ) with S = √ . In particular, ( X, d ) has Enflotype with E ( X ) ≤ √ . roof. For any four points w, x, y, z ∈ X , the Ptolemy inequality (5.6) yields d ( w, y ) + d ( x, z ) − { d ( w, y ) − d ( x, z ) } = 2 d ( w, y ) · d ( x, z ) ≤ d ( w, x ) · d ( y, z ) + 2 d ( w, z ) · d ( y, x ) ≤ d ( w, x ) + d ( y, z ) + d ( w, z ) + d ( y, x ) . It follows from the triangle inequality that { d ( w, y ) − d ( x, z ) } ≤ { d ( w, x ) + d ( y, z ) } ≤ d ( w, x ) + 2 d ( y, z ) . Therefore we obtain d ( w, y ) + d ( x, z ) ≤ d ( w, x ) + d ( y, z ) + d ( w, z ) + d ( y, x ) + { d ( w, y ) − d ( x, z ) } ≤ { d ( w, x ) + d ( y, z ) } + d ( w, z ) + d ( y, x ) . The proof of Proposition 5.2 shows that this inequality derives Enflo type 2. ✷ References [B] K. Ball,
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