## Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces

Citations: | 39 - 23 self |

### BibTeX

@MISC{Naor_markovchains,

author = {Assaf Naor and Yuval Peres and Oded Schramm and Scott Sheffield},

title = {Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

A metric space X has Markov type 2, if for any reversible finite-state Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D 2 t) ≤ K 2 t E(D 2 1) for some K = K(X) < ∞. This notion is due to K. Ball (1992), who showed its importance for the Lipschitz extension problem. However until now, only Hilbert space (and its bi-Lipschitz equivalents) were known to have Markov type 2. We show that every Banach space with modulus of smoothness of power type 2 (in particular, Lp for p> 2) has Markov type 2; this proves a conjecture of Ball. We also show that trees, hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature have Markov type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in 1982, by showing that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension defined on all of Lp. 1

### Citations

511 |
Metric Structures for Riemannian and Non-Riemannian Spaces (Birkhäuser
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(Show Context)
Citation Context ...lt which holds for arbitrary Gromov hyperbolic spaces. One of many alternative definitions for Gromov-hyperbolic spaces is as follows (background material on this topic can be found in the monographs =-=[23, 8, 24, 64]-=-). Let (X, d) be a metric space. For x, y, r ∈ X the Gromov product with respect to r is defined as: 〈x|y〉r := d(x, r) + d(y, r) − d(x, y) . (1) 2 For δ ≥ 0, the metric space X is said to be δ-hyperbo... |

404 |
Extensions of Lipschitz mapping into hilbert space
- Johnson, Lindenstrauss
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(Show Context)
Citation Context ...conjunction with Ball’s extension theorem [2], this implies the following non-linear version of Maurey’s extension theorem [49] (see below), answering a question posed by Johnson and Lindenstrauss in =-=[29]-=-. Theorem 1.3. Let X be a Banach space with modulus of smoothness of power type 2 and let Y be a Banach space with modulus of convexity of power type 2. Then there exists a constant C = C(X, Y ) such ... |

359 |
Metric Spaces of Non-Positive Curvature
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(Show Context)
Citation Context ...lt which holds for arbitrary Gromov hyperbolic spaces. One of many alternative definitions for Gromov-hyperbolic spaces is as follows (background material on this topic can be found in the monographs =-=[23, 8, 24, 64]-=-). Let (X, d) be a metric space. For x, y, r ∈ X the Gromov product with respect to r is defined as: 〈x|y〉r := d(x, r) + d(y, r) − d(x, y) . (1) 2 For δ ≥ 0, the metric space X is said to be δ-hyperbo... |

282 |
The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts
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(Show Context)
Citation Context ...tric and analytic properties of a normed space X, and in the past three decades the theory of type and cotype has developed into a deep and rich theory. We refer to the survey article [51], the books =-=[44, 53, 60, 66, 14]-=- and the references therein for numerous results and techniques in this direction. A fundamental result of Maurey [49] states that any bounded linear operator from a linear subspace of a Banach space ... |

195 |
Geometric nonlinear functional analysis
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(Show Context)
Citation Context ...ing � ˜ f�Lip ≤ K�f�Lip (if no such K exists we set e(X, Y ) = ∞). Estimating e(X, Y ) under various geometric conditions on X and Y is a classical problem which dates back to the 1930’s. We refer to =-=[65, 6, 41]-=- and the references therein for an account of known results on Lipschitz extension. The modern theory of the Lipschitz extension problem between normed spaces starts with the work of Marcus-Pisier [47... |

169 |
Asymptotic theory of finite dimensional normed spaces
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(Show Context)
Citation Context ...tric and analytic properties of a normed space X, and in the past three decades the theory of type and cotype has developed into a deep and rich theory. We refer to the survey article [51], the books =-=[44, 53, 60, 66, 14]-=- and the references therein for numerous results and techniques in this direction. A fundamental result of Maurey [49] states that any bounded linear operator from a linear subspace of a Banach space ... |

156 |
Ebin – Comparison theorems in Riemannian geometry
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Citation Context ...t X is δ hyperbolic (with δ proportional to 1/r). The fact that there exists ε > 0 such that all the balls of of radius ε in X embed bi-Lipschitzly in R n follows from Rauch’s comparison theorem (see =-=[12]-=- and Chapter 8+ in [24]). Thus the required result follows from Theorem 6.4. 18sG0 G1 G2 G3 Figure 1: The Laakso graphs. 7 The Laakso graphs, doubling spaces and weak Markov type Recall that a metric ... |

118 | Lectures on Analysis on Metric Spaces - Heinonen - 2001 |

101 |
trees and degenerations of hyperbolic structures
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(Show Context)
Citation Context ... there is a path in X from x to y whose length is d(x, y). An R-tree is a path metric space (T, d) such that for every two distinct points x, y ∈ T there is a unique simple path from x to y in T (see =-=[13, 54]-=-). (Some definitions appearing in the literature also require the metric to be complete.) Equivalently, an R-tree is a 0-hyperbolic metric space whose metric is a path metric. An r-separated set A in ... |

89 | Probability: theory and examples, Second edition - Durrett - 1996 |

83 | Measured descent: A new embedding method for finite metrics
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(Show Context)
Citation Context ...ζ) ≤ Pr dY (g√ tζ (f(Z0)), g√ tζ (f(Zt))) 2 ≥ tζ K2 � ≤ M w 2 (Y ) 2 K 2 ζ −1 � E dY (g√ tζ (f(Z0)), g√ tζ (f(Zt))) 2� ≤ M w 2 (Y ) 2 K 2 ζ −1 E � dX(f(Z0), f(Zt)) 2� . It follows from the results of =-=[42, 35]-=- that if X is doubling with constant λ then X embeds weakly into Hilbert space with distortion O(log λ). This yields an alternative proof of Theorem 7.3, with the concrete estimate M w 2 (X) = O(log λ... |

79 |
Plongements lipschitziens dans R n
- Assouad
- 1983
(Show Context)
Citation Context ...mposition (note that A is self-adjoint on L 2 (π)). The proof of the following lemma is a slight modification of the proof of Lemma 3.1 in [2]. Let Lq(X) denote the collection of Borel measurable f : =-=[0, 1]-=- → X with E�f� q = � 1 0 �f�q < ∞. It is a Banach space with norm � E�f� q� 1/q . (Of course, the choice of the interval [0, 1] as the domain for f is rather arbitrary. It may be replaced with any pro... |

79 |
Series de variables aleatoires vectorielles independantes et proprietes geometriques des espaces de banach
- Maurey, Pisier
- 1976
(Show Context)
Citation Context ...xi� 2 encodes the rich geometric structure of Hilbert space. In the early 1970’s, the work of Dubinsky-Pe̷lczyńsky-Rosenthal [15], Hoffmann-Jørgensen [27], Kwapien [36], Maurey [48] and Maurey-Pisier =-=[52]-=- has led to the notions of (Rademacher) type and cotype, which are natural relaxations of the generalized parallelogram identity. A Banach space X is said to have type p > 0 if there exists a constant... |

78 |
Absolutely Summing Operators
- Diestel, Jarchow, et al.
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(Show Context)
Citation Context ...tric and analytic properties of a normed space X, and in the past three decades the theory of type and cotype has developed into a deep and rich theory. We refer to the survey article [51], the books =-=[44, 53, 60, 66, 14]-=- and the references therein for numerous results and techniques in this direction. A fundamental result of Maurey [49] states that any bounded linear operator from a linear subspace of a Banach space ... |

72 |
la Harpe (eds), Sur les groupes hyperboliques d’apres de Mikhael Gromov, Birkhäuser
- Ghys, de
- 1990
(Show Context)
Citation Context ...lt which holds for arbitrary Gromov hyperbolic spaces. One of many alternative definitions for Gromov-hyperbolic spaces is as follows (background material on this topic can be found in the monographs =-=[23, 8, 24, 64]-=-). Let (X, d) be a metric space. For x, y, r ∈ X the Gromov product with respect to r is defined as: 〈x|y〉r := d(x, r) + d(y, r) − d(x, y) . (1) 2 For δ ≥ 0, the metric space X is said to be δ-hyperbo... |

72 | Martingales with values in uniformly convex spaces - Pisier |

68 |
Probabilistic methods in the geometry of Banach spaces. Probability and Analysis, Varenna (Italy
- Pisier
- 1985
(Show Context)
Citation Context ...type p — this follows by considering cubes of the form xε = �n i=1 εi yi, where y1, . . . , yn ∈ X. A variant of Enflo type was introduced and studied by Bourgain, Milman and Wolfson in [9] (see also =-=[59]-=-). In [56], it was shown that for a wide class of normed spaces, Rademacher type p implies Enflo type p. Despite 4 ε∼ε ′ i=1sthe usefulness of these notions of non-linear type to various embedding pro... |

66 | On metric Ramsey-type phenomena
- Bartal, Linial, et al.
- 2004
(Show Context)
Citation Context ...troduced this concept in his profound study of the Lipschitz extension problem [2] (see Section 2), and the notion of Markov type has since found applications in the theory of bi-Lipschitz embeddings =-=[46, 4]-=-. The main theorem in [2] states that Lipschitz functions from a subset of a metric space X having Markov type 2 into a Banach space with modulus of convexity of power type 2 (see the definition in (3... |

63 | Embedding the diamond graph in lp and dimension reduction
- Lee, Naor
- 2004
(Show Context)
Citation Context ...is the following. In [42] it was shown that if {0, 1} d embeds into ℓ k ∞ with distortion D then k ≥ 2 Ω(d/D2 ) . This fact easily follows from the above discussion via an argument similar to that of =-=[40]-=-: By Hölder’s inequality ℓ k ∞ and ℓ k p are O(1) equivalent when p = log k. Thus, the fact that {0, 1} d embeds into ℓ k ∞ with distortion D implies that D ≥ Ω( � d/p) = Ω( � d/ log k), which simplif... |

50 |
Markov chains, Riesz transforms and Lipschitz maps
- Ball
- 1992
(Show Context)
Citation Context ...sed by Johnson and Lindenstrauss in 1982, by showing that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension defined on all of Lp. 1 Introduction K. Ball =-=[2]-=- introduced the notion of Markov type of metric spaces, defined as follows. Recall that a Markov chain {Zt} ∞ t=0 with transition probabilities aij := Pr(Zt+1 = j | Zt = i) on the state space {1, . . ... |

49 | Embeddings of Gromov hyperbolic spaces
- Bonk, Schramm
(Show Context)
Citation Context ...us ε embed in R n with distortion at most D, there is an upper bound (depending only on n and D) for the cardinality of any ε/2-separated set in X whose diameter is less than ε. In the terminology of =-=[7]-=-, this means that X has bounded growth in some scale. By a theorem of Bonk and Schramm [7], there exists an integer m such that X is quasi-isometric to a subset of the m dimensional hyperbolic space H... |

46 |
uniform convexity and smoothness inequalities for trace norms
- Ball, Carlen, et al.
(Show Context)
Citation Context ...s said to have modulus of convexity of power type q if there exists a constant c such that δ(ε) ≥ c ε q for all ε ∈ [0, 2]. It is straightforward to check that in this case q ≥ 2. By Proposition 7 in =-=[3]-=- (see also [21]), X has modulus of convexity of power type q if and only if there exists a constant K > 0 such that for every x, y ∈ X 2 �x� q + 2 K q �y�q ≤ �x + y� q + �x − y� q . (4) The least K fo... |

46 |
Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality
- Laakso
(Show Context)
Citation Context ...ces cannot go through embeddings in Hilbert space. The standard example showing that in both embedding results we must pass to a power of the metric is the family of graphs known as the Laakso graphs =-=[37, 38]-=-, which are planar graphs whose graph metric is uniformly doubling, yet they do not uniformly embed into Hilbert space. These graphs Gk are defined recursively as in Figure 1. Proposition 7.1. The Laa... |

44 |
On type of metric spaces
- Bourgain, Milman, et al.
- 1986
(Show Context)
Citation Context ...es Rademacher type p — this follows by considering cubes of the form xε = �n i=1 εi yi, where y1, . . . , yn ∈ X. A variant of Enflo type was introduced and studied by Bourgain, Milman and Wolfson in =-=[9]-=- (see also [59]). In [56], it was shown that for a wide class of normed spaces, Rademacher type p implies Enflo type p. Despite 4 ε∼ε ′ i=1sthe usefulness of these notions of non-linear type to variou... |

43 | Extending Lipschitz functions via random metric partitions
- Lee, Naor
(Show Context)
Citation Context ...ing � ˜ f�Lip ≤ K�f�Lip (if no such K exists we set e(X, Y ) = ∞). Estimating e(X, Y ) under various geometric conditions on X and Y is a classical problem which dates back to the 1930’s. We refer to =-=[65, 6, 41]-=- and the references therein for an account of known results on Lipschitz extension. The modern theory of the Lipschitz extension problem between normed spaces starts with the work of Marcus-Pisier [47... |

39 |
Holomorphic semi-groups and the geometry of Banach spaces
- Pisier
- 1982
(Show Context)
Citation Context ...be that for every Banach space, Rademacher type p implies Enflo type p, or even Markov type p. This amounts to proving that for Banach spaces of type greater than 1 (also known as K-convex space. See =-=[58]-=- for the geometric and analytic ramifications of this assumption), the Rademacher type and Enflo type (or Markov type) coincide. One simple example of a class of spaces for which we can prove that the... |

33 |
Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces L p . Société Mathématique de
- Maurey
- 1974
(Show Context)
Citation Context ...Markov type 2. Moreover, for every 2 ≤ p < ∞, we have M2(Lp) ≤ 4 √ p − 1. In conjunction with Ball’s extension theorem [2], this implies the following non-linear version of Maurey’s extension theorem =-=[49]-=- (see below), answering a question posed by Johnson and Lindenstrauss in [29]. Theorem 1.3. Let X be a Banach space with modulus of smoothness of power type 2 and let Y be a Banach space with modulus ... |

32 |
Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes
- Marcus, Pisier
- 1984
(Show Context)
Citation Context ...41] and the references therein for an account of known results on Lipschitz extension. The modern theory of the Lipschitz extension problem between normed spaces starts with the work of Marcus-Pisier =-=[47]-=- and Johnson-Lindenstrauss [29]. In [29] it is asked if there is a nonlinear analog of Maurey’s extension theorem. To investigate this question it is clearly necessary to develop non-linear variants o... |

31 |
Bilipschitz embeddings of metric spaces into space forms
- Lang, Plaut
- 2001
(Show Context)
Citation Context ...ces cannot go through embeddings in Hilbert space. The standard example showing that in both embedding results we must pass to a power of the metric is the family of graphs known as the Laakso graphs =-=[37, 38]-=-, which are planar graphs whose graph metric is uniformly doubling, yet they do not uniformly embed into Hilbert space. These graphs Gk are defined recursively as in Figure 1. Proposition 7.1. The Laa... |

28 |
Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients
- Kwapien
- 1972
(Show Context)
Citation Context ...} �ε1x1 + . . . + εnxn�2 = � n i=1 �xi� 2 encodes the rich geometric structure of Hilbert space. In the early 1970’s, the work of Dubinsky-Pe̷lczyńsky-Rosenthal [15], Hoffmann-Jørgensen [27], Kwapien =-=[36]-=-, Maurey [48] and Maurey-Pisier [52] has led to the notions of (Rademacher) type and cotype, which are natural relaxations of the generalized parallelogram identity. A Banach space X is said to have t... |

28 | Metric structures in L1: dimension, snowflakes, and average distortion
- Lee, Mendel, et al.
- 2005
(Show Context)
Citation Context ...ere c is a universal constant. Thus the embedding x ↦→ Zx incurs distortion Ω( � d/p). An application of having good bounds in terms of p on the Lp distortion of the Hamming cube is the following. In =-=[42]-=- it was shown that if {0, 1} d embeds into ℓ k ∞ with distortion D then k ≥ 2 Ω(d/D2 ) . This fact easily follows from the above discussion via an argument similar to that of [40]: By Hölder’s inequal... |

27 | Remarks on non linear type and Pisier’s inequality
- Naor, Schechtman
(Show Context)
Citation Context ...ll be discussed later in this introduction. Our methods yield Markov type 2 for several new classes of spaces; in particular we have the following result, which answers positively a question posed in =-=[56]-=-. Theorem 1.4. There exists a universal constant C > 0 such that for every tree T with arbitrary positive edge lengths, M2(T ) ≤ C. In this theorem, as well as in Theorem 1.5 and Corollary 1.6 below, ... |

26 |
Martingales and singular integrals in Banach spaces
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- 2001
(Show Context)
Citation Context ...every p > 1, if a Banach space X has Rademacher type p then it also has Enflo type q for every q < p. No such result is known for Markov type. In [56] it is shown that if X is a UMD Banach space (see =-=[10]-=- for details on UMD spaces) of Rademacher type p, then X also has Enflo type p. It would be desirable to obtain a result stating that for a certain class of Banach spaces, the notions of Rademacher ty... |

26 |
On the moduli of convexity and smoothness
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- 1976
(Show Context)
Citation Context ... modulus of convexity of power type q if there exists a constant c such that δ(ε) ≥ c ε q for all ε ∈ [0, 2]. It is straightforward to check that in this case q ≥ 2. By Proposition 7 in [3] (see also =-=[21]-=-), X has modulus of convexity of power type q if and only if there exists a constant K > 0 such that for every x, y ∈ X 2 �x� q + 2 K q �y�q ≤ �x + y� q + �x − y� q . (4) The least K for which (4) hol... |

24 |
Banach Spaces for Analysts
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(Show Context)
Citation Context |

22 |
Über die zusammenziehenden und Lipschitzchen Transformationen
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(Show Context)
Citation Context ...em 2.4 is redundant. In particular it seems likely that for 2 ≤ p < ∞ and 1 < q ≤ 2, e(Lp, Lq) ≤ � (p − 1)/(q − 1). If true, this would be a generalization of Kirszbraun’s classical extension theorem =-=[34]-=- (see also [65, 6]). 5. Since L1 has cotype 2 but isn’t uniformly convex, there is no known non-linear analog of Maurey’s extension theorem for L1-valued mappings. In particular, it isn’t known whethe... |

16 |
On infinite-dimensional topological groups, Séminaire sur la Géométrie des Espaces de Banach
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- 1977
(Show Context)
Citation Context ...in the past 20 years on non-linear notions of type, a satisfactory notion of non-linear cotype remains elusive. Enflo [17, 18, 19] studied the notion of roundness of metric spaces and subsequently in =-=[20]-=-, generalized roundness to a notion which is known today as Enflo type. Let X be a metric space and fix n ∈ N. An n-dimensional cube in X is a mapping ε ↦→ xε from {−1, 1} n to X. X is said to have En... |

16 |
On the uniform convexity of L p and l p
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(Show Context)
Citation Context ...and is denoted Sp(X). It was shown in [3] (see also [21]) that K2(Lp) ≤ 1/ √ p − 1 for 1 < p ≤ 2, and S2(Lp) ≤ √ p − 1 for 2 ≤ p < ∞ (the order of magnitude of these constants was first calculated in =-=[25]-=-). In [22, 21] (see also [44], Theorem 1.e.16.) it is shown that if a Banach space X has modulus of convexity of power type q then X also has cotype q. Similarly, if X has modulus of smoothness of pow... |

16 |
Nonreflexive spaces of type 2
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- 1978
(Show Context)
Citation Context ... of smoothness of power type p then X has type p. Observe that L1 has cotype 2 (see [44, 53]), but it is clearly not uniformly convex. There also exist spaces of type 2 which are not uniformly smooth =-=[28, 61]-=-, but these spaces are much harder to construct. For all the classical reflexive spaces, the power type of smoothness and convexity coincide with their type and cotype, respectively. Thus, in the cont... |

16 |
On inner products in linear metric spaces
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- 1935
(Show Context)
Citation Context ...ak form of Markov type 2. Finally, Section 8 contains some open problems. 2 Linear and non-linear type and cotype; the linear and Lipschitz extension problems The classical Jordan-von Neumann theorem =-=[31]-=- states that Hilbert space is characterized among Banach spaces by the parallelogram identity �x + y�2 + �x − y�2 = 2�x�2 + 2�y�2 � . Inductively it follows that the generalized parallelogram identity... |

15 |
On the modulus of smoothness and divergent series in Banach spaces
- Lindenstrauss
- 1963
(Show Context)
Citation Context ... of elements x1, x2, . . . in a Banach space Y is called a monotone basic sequence if for every a1, a2, . . . ∈ R and every integer n, � � � n� i=1 aixi � �n+1 � �� � ≤ � By a result of Lindenstrauss =-=[43]-=- (see also [57, Proposition 2.2]), for a Banach space Y , if ρY (τ) ≤ Kτ q for all τ, then for every monotone basic sequence {xi}i≥0 in Y , � � � n� i=1 � � xi� q ≤ 4 q K i=1 aixi � � �. n� �xi� q . L... |

15 |
Embeddings and Extensions in Analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84
- Wells, Williams
- 1982
(Show Context)
Citation Context ...ing � ˜ f�Lip ≤ K�f�Lip (if no such K exists we set e(X, Y ) = ∞). Estimating e(X, Y ) under various geometric conditions on X and Y is a classical problem which dates back to the 1930’s. We refer to =-=[65, 6, 41]-=- and the references therein for an account of known results on Lipschitz extension. The modern theory of the Lipschitz extension problem between normed spaces starts with the work of Marcus-Pisier [47... |

13 | Girth and Euclidean distortion
- Linial, Magen, et al.
(Show Context)
Citation Context ...troduced this concept in his profound study of the Lipschitz extension problem [2] (see Section 2), and the notion of Markov type has since found applications in the theory of bi-Lipschitz embeddings =-=[46, 4]-=-. The main theorem in [2] states that Lipschitz functions from a subset of a metric space X having Markov type 2 into a Banach space with modulus of convexity of power type 2 (see the definition in (3... |

13 |
Type, cotype and K-convexity. Handbook of the geometry of Banach spaces
- Maurey
- 2003
(Show Context)
Citation Context ...ng list of geometric and analytic properties of a normed space X, and in the past three decades the theory of type and cotype has developed into a deep and rich theory. We refer to the survey article =-=[51]-=-, the books [44, 53, 60, 66, 14] and the references therein for numerous results and techniques in this direction. A fundamental result of Maurey [49] states that any bounded linear operator from a li... |

13 |
A phase transition phenomenon between the isometric and isomorphic extension problems for Hölder functions between Lp spaces
- Naor
- 2003
(Show Context)
Citation Context ...rem: Theorem 2.4. For every two Banach spaces X, Y , e(X, Y ) ≤ 24 S2(X) K2(Y ). In particular, for 2 ≤ p < ∞ and 1 < q ≤ 2, � p − 1 e(Lp, Lq) ≤ 24 q − 1 . 3 Some additional geometric applications In =-=[55]-=- the extension problem for Hölder functions was studied. Let X, Y be metric spaces. Recall that a function f : X → Y is α Hölder with constant K if for every x, y ∈ X, d(f(x), f(y)) ≤ Kd(x, y) α . Fol... |

13 |
Random series in the real interpolation spaces between the spaces vp
- Pisier, Xu
- 1987
(Show Context)
Citation Context ... of smoothness of power type p then X has type p. Observe that L1 has cotype 2 (see [44, 53]), but it is clearly not uniformly convex. There also exist spaces of type 2 which are not uniformly smooth =-=[28, 61]-=-, but these spaces are much harder to construct. For all the classical reflexive spaces, the power type of smoothness and convexity coincide with their type and cotype, respectively. Thus, in the cont... |

11 |
Uniform structures and square roots in topological groups
- Enflo
- 1970
(Show Context)
Citation Context ...non-linear variants of type and cotype. While there has been substantial progress in the past 20 years on non-linear notions of type, a satisfactory notion of non-linear cotype remains elusive. Enflo =-=[17, 18, 19]-=- studied the notion of roundness of metric spaces and subsequently in [20], generalized roundness to a notion which is known today as Enflo type. Let X be a metric space and fix n ∈ N. An n-dimensiona... |

10 | Lipschitz quotients from metric trees and from Banach spaces containing ℓ1 - Johnson, Lindenstrauss, et al. |

10 |
Type et cotype dans les espaces munis de structures locales inconditionnelles
- Maurey
- 1974
(Show Context)
Citation Context ...n and sequence spaces, with the point-wise partial order. We refer to [44] for an account of the beautiful theory of Banach lattices. A combination of a theorem of Figiel [21] and a theorem of Maurey =-=[50]-=- (see Theorem 1.f.1. and Proposition 1.f.17. in [44]) implies that a Banach lattice X of type 2 can be renormed to have a modulus of smoothness of power type 2. Thus by Theorem 2.3 X has Markov type 2... |

9 |
Sums of independent Banach space valued random variables
- Hoffman-Jørgensen
- 1974
(Show Context)
Citation Context ... 2−n εi∈{−1,+1} �ε1x1 + . . . + εnxn�2 = � n i=1 �xi� 2 encodes the rich geometric structure of Hilbert space. In the early 1970’s, the work of Dubinsky-Pe̷lczyńsky-Rosenthal [15], Hoffmann-Jørgensen =-=[27]-=-, Kwapien [36], Maurey [48] and Maurey-Pisier [52] has led to the notions of (Rademacher) type and cotype, which are natural relaxations of the generalized parallelogram identity. A Banach space X is ... |