## Metric cotype (2005)

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Citations: | 27 - 16 self |

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@TECHREPORT{Mendel05metriccotype,

author = {Manor Mendel and Assaf Naor},

title = {Metric cotype},

institution = {},

year = {2005}

}

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### Abstract

We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either almost universal (i.e., contains any finite metric space with any distortion> 1), or there exists α> 0, and arbitrarily large n-point metrics whose distortion when embedded in any member of F is at least Ω ((log n) α). The same property is also used to prove strong non-embeddability theorems of Lq into Lp, when q> max{2, p}. Finally we use metric cotype to obtain a new type of isoperimetric inequality on the discrete torus. 1

### Citations

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Citation Context ...non-trivial (and non-linear) interactions between cuts. Strong nonembeddability results for Lp. To state these results we need the following weak notion of distance respecting embedding due to Gromov =-=[18]-=-. DEFINITION 1.4. Let (M, dM) and (N , dN ) be metric spaces. A mapping f : M → N is called a coarse embedding if there exists two non-decreasing functions α, β : [0, ∞) → [0, ∞) such that limt→∞ α(t)... |

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Citation Context ... this notion in what follows. i.e.2. there exists α > 0 and a sequence of metric spaces {Mn}n≥1, such that |Mn| = n, and cF(Mn) = Ω ((log n) α ). For Hilbert space H, sup{cH(M) : |M| = n} = Θ(log n) =-=[8, 26]-=-. We do not know whether there exists a class of metric spaces F which is not almost universal, but for which sup{cF(M) : |M| = n} = O ( (log n) β) , for some β ∈ (0, 1). Theorem 1.2 is proved using a... |

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Citation Context ...ithm design. One approach in this vein is to reduce optimization problems over general metric spaces to a class of “special” metrics which has more structure (e.g., convex combination of tree metrics =-=[1, 5]-=-), and solve the optimization problem over the class of special metrics. The class of special metric spaces is chosen to balance between the structure needed for developing an algorithmic solution, an... |

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Citation Context ...n University of Israel, mendelma@gmail.com ‡ Microsoft Research, anaor@microsoft.com The smallest such distortion is denoted cN (M), cN (M) := inf{dist(f) : f : M ↩→ N }. Bourgain’s embedding theorem =-=[8]-=- and Bartal’s probabilistic embedding theorem [5, 16] established Hilbert spaces, ℓ1, and and convex combination of tree metrics as useful host spaces for which the distortion of embedding npoint metr... |

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Citation Context ...osoft Research, anaor@microsoft.com The smallest such distortion is denoted cN (M), cN (M) := inf{dist(f) : f : M ↩→ N }. Bourgain’s embedding theorem [8] and Bartal’s probabilistic embedding theorem =-=[5, 16]-=- established Hilbert spaces, ℓ1, and and convex combination of tree metrics as useful host spaces for which the distortion of embedding npoint metrics is O(log n). It is therefore interesting to find ... |

207 |
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Citation Context ...ove. For perspective, we mention some previous examples of this interaction between Banach space theory and computer science: I. Bourgain’s famous embedding theorem [8] is motivated by John’s theorem =-=[22]-=-. Bourgain’s embedding technique has found many applications in computer science (see [21]). II. Bourgain’s work on the metric interpretation of superreflexivity [9] has been followed up by computer s... |

195 |
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(Show Context)
Citation Context ...omorphic to ℓd 2, and d = ΩqX ,η(n2/qX ). The notions of type and cotype are clearly linear notions, since their definition involves addition and multiplication by scalars. However, in 1976 Ribe (see =-=[7]-=-) proved that if X and Y are uniformly homeomorphic Banach spaces (i.e., there exists a bijection f which is uniformly continuous and f −1 is also uniformly continuous) then X is finitely representabl... |

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Citation Context ...ithm design. One approach in this vein is to reduce optimization problems over general metric spaces to a class of “special” metrics which has more structure (e.g., convex combination of tree metrics =-=[1, 5]-=-), and solve the optimization problem over the class of special metrics. The class of special metric spaces is chosen to balance between the structure needed for developing an algorithmic solution, an... |

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Citation Context ...ilbert space if and only if it has type 2 and cotype 2. 2. Denote by pX the supremum over p such that X has type p, and by by qX the infimum over q such that X has cotype q. The Maurey-Pisier theorem =-=[35, 32]-=- states that for any n ∈ N, and any η > 0, X linearly contains copies of ℓn qX and ℓnpX with distortion at most 1 + η. 3. Dvoretzky’s theorem (see [31, Chap. 14]) states that for any η > 0 and n ∈ N, ... |

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Citation Context ...ar subspace Y ⊆ X that is isomorphic to ℓ d 2 with distortion 1 + η, where d = Ωη(log n). The logarithmic estimate on d is known to be asymptotically tight. However, Figiel, Lindenstrauss, and Milman =-=[17]-=- have shown that it is possible find such Y which is 1 + η isomorphic to ℓd 2, and d = ΩqX ,η(n2/qX ). The notions of type and cotype are clearly linear notions, since their definition involves additi... |

68 |
Probabilistic methods in the geometry of Banach spaces. Probability and Analysis, Varenna (Italy
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(Show Context)
Citation Context ... it remains open. Secondly, we do not know if (1.4) is a useful notion, in the sense that it yields metric variants of certain theorems from the linear theory of type. The first issue is addressed in =-=[10, 36]-=- where it is shown that for Banach spaces, Rademacher type p implies Enflo type p ′ for every 0 < p ′ < p, and the same holds for a variant of Enflo type called BMW type. The second issue turned out n... |

66 | On metric Ramsey-type phenomena
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- 2004
(Show Context)
Citation Context ...Markov type, was introduced by Ball [4] in his study of the Lipschitz extension problem. This important notion has since found applications to various problems in metric geometry and computer science =-=[27, 6, 34]-=-. Despite the vast amount of research on non-linear type, a non-linear notion of cotype remained elusive. Indeed, the problem of finding a notion of cotype which makes sense for arbitrary metric space... |

65 | Lectures on discrete geometry, volume 212 of Graduate Texts in Mathematics - Matouˇsek - 2002 |

65 | The Coarse Baum-Connes Conjecture for Spaces Which Admit a Uniform Embedding into Hilbert Space,to appear - Yu |

62 |
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- Bourgain
- 1986
(Show Context)
Citation Context ...ns of this phenomenon for specific local properties of Banach spaces (such as type, cotype and super-reflexivity), has long been a major driving force in the bi-Lipschitz theory of metric spaces (see =-=[9]-=- for a discussion of this research program). Once this is achieved, one could define the notion of type and cotype of a metric space, and then hopefully transfer some of the deep theory of type and co... |

50 |
Markov chains, Riesz transforms and Lipschitz maps
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(Show Context)
Citation Context ...mentioned above, yielding a characterization of metric spaces which contain bi-Lipschitz copies of the Hamming cube. A stronger notion of non-linear type, known as Markov type, was introduced by Ball =-=[4]-=- in his study of the Lipschitz extension problem. This important notion has since found applications to various problems in metric geometry and computer science [27, 6, 34]. Despite the vast amount of... |

50 |
Geometry of cuts and metrics, volume 15 of Algorithms and Combinatorics
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(Show Context)
Citation Context ...of our knowledge, all the known non-embeddability results for L1 are based on Poincaré type inequalities in which distances are raised to the power 1. By the cutcone representation of L1 metrics (see =-=[11]-=-) it is enough to prove any such inequality for cut metrics, which are particularly simple. Theorem 1.3 seems to be the first truly “infinite dimensional” metric inequality in L1. We believe that unde... |

49 | Low-distortion embeddings of finite metric spaces
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- 2004
(Show Context)
Citation Context ... space theory and computer science: I. Bourgain’s famous embedding theorem [8] is motivated by John’s theorem [22]. Bourgain’s embedding technique has found many applications in computer science (see =-=[21]-=-). II. Bourgain’s work on the metric interpretation of superreflexivity [9] has been followed up by computer scientists regarding the embeddability of of tree metrics in Euclidean spaces [19, 28, 30, ... |

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44 |
On type of metric spaces
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(Show Context)
Citation Context ... it remains open. Secondly, we do not know if (1.4) is a useful notion, in the sense that it yields metric variants of certain theorems from the linear theory of type. The first issue is addressed in =-=[10, 36]-=- where it is shown that for Banach spaces, Rademacher type p implies Enflo type p ′ for every 0 < p ′ < p, and the same holds for a variant of Enflo type called BMW type. The second issue turned out n... |

44 | Probability in Banach spaces, volume 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3 - Ledoux, Talagrand - 1991 |

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39 | Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces
- Naor, Peres, et al.
- 2006
(Show Context)
Citation Context ...Markov type, was introduced by Ball [4] in his study of the Lipschitz extension problem. This important notion has since found applications to various problems in metric geometry and computer science =-=[27, 6, 34]-=-. Despite the vast amount of research on non-linear type, a non-linear notion of cotype remained elusive. Indeed, the problem of finding a notion of cotype which makes sense for arbitrary metric space... |

39 | Holomorphic semi-groups and the geometry of Banach spaces - Pisier - 1982 |

38 |
On the nonexistence of uniform homeomorphisms between Lp-spaces
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- 1969
(Show Context)
Citation Context ...could define the notion of type and cotype of a metric space, and then hopefully transfer some of the deep theory of type and cotype to the context of arbitrary metric spaces. Enflo’s pioneering work =-=[12, 13, 14, 15]-=- resulted in the formulation of a non-linear notion of type, known today as Enflo type. The basic idea is that given a Banach space X and x1, . . . , xn ∈ X, one can consider the linear function f : {... |

35 |
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Citation Context ...er C[0, 1] is uniformly homeomorphic to a subset of M (the space C[0, 1] can be replaced here by c0 due to Aharoni’s theorem [1]). Enflo proved that c0 does not uniformly embed into Hilbert space. In =-=[2]-=-, Aharoni, Maurey and Mityagin systematically studied metric spaces which are uniformly homeomorphic to a subset of Hilbert space, and obtained an elegant characterization of Banach spaces which are u... |

32 |
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Citation Context ...m 1.6 generalizes a recent result of Johnson and Randrianarivony [23] who proved a special case of Theorem 1.6 when p ∈ [1, 2]. This completes the coarse classification of Lp spaces since it is known =-=[37, 33]-=- that Lq coarsely embeds in Lp when q ≤ p or when q ≤ 2. Similar results hold for another type of weak embedding called uniform embedding. We will not discuss this topic here, and refer to the full ve... |

30 | On bases, finite dimensional decompositions and weaker structures in Banach spaces - Johnson, Rosenthal, et al. - 1971 |

28 |
Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients
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Citation Context ...verse aspects of the local theory of Banach spaces. We refer to the full version of this paper for references on these topics. Here we mention only few highlights of this theory: 1. Kwapien’s Theorem =-=[24]-=- generalizes the isometric characterization of Hilbert space into an isomorphic one: A Banach space X is isomorphic (i.e., has a linear bijection with finite distortion) to Hilbert space if and only i... |

28 | Euclidean quotients of finite metric spaces
- Mendel, Naor
- 2004
(Show Context)
Citation Context ...m 1.6 generalizes a recent result of Johnson and Randrianarivony [23] who proved a special case of Theorem 1.6 when p ∈ [1, 2]. This completes the coarse classification of Lp spaces since it is known =-=[37, 33]-=- that Lq coarsely embeds in Lp when q ≤ p or when q ≤ 2. Similar results hold for another type of weak embedding called uniform embedding. We will not discuss this topic here, and refer to the full ve... |

27 | Remarks on non linear type and Pisier’s inequality - Naor, Schechtman |

26 | Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces - Heinrich, Mankiewicz - 1982 |

26 | The moduli of smoothness and convexity and the Rademacher averages of trace classes - Tomczak-Jaegermann - 1974 |

25 |
On embedding trees into uniformly convex Banach spaces
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(Show Context)
Citation Context ...nce (see [21]). II. Bourgain’s work on the metric interpretation of superreflexivity [9] has been followed up by computer scientists regarding the embeddability of of tree metrics in Euclidean spaces =-=[19, 28, 30, 20, 25]-=-. III. Ball’s notion of Markov type [4], partially motivated as another metric analog for type, has been used by the computer science community to analyze the Euclidean distortion of high-girth graphs... |

24 | On Hilbertian subsets of finite metric spaces - Bourgain, Figiel, et al. - 1986 |

23 | Some results on Banach spaces without local unconditional structure - Pisier - 1978 |

23 | Probability in Banach spaces Ergebnisse der Mathematik - Ledoux, Talagrand - 1991 |

20 |
Cuts, trees and l1-embeddings of graphs
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(Show Context)
Citation Context ...nce (see [21]). II. Bourgain’s work on the metric interpretation of superreflexivity [9] has been followed up by computer scientists regarding the embeddability of of tree metrics in Euclidean spaces =-=[19, 28, 30, 20, 25]-=-. III. Ball’s notion of Markov type [4], partially motivated as another metric analog for type, has been used by the computer science community to analyze the Euclidean distortion of high-girth graphs... |