Every separable metric space is Lipschitz equivalent to a subset ofc 0 +

Aharoni, Israel

doi:10.1007/bf02757727

Cited by 74 publications

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“…Smirnov's problem was settled negatively by Enflo in [17]. Following Enflo, we shall say that a metric space M is a universal uniform embedding space if every separable metric space embeds uniformly into M. Since every separable metric space is isometric to a subset of C[0, 1], this is equivalent to asking whether C[0, 1] is uniformly homeomorphic to a subset of M (the space C[0, 1] can be replaced here by c 0 due to Aharoni's theorem [1]). Enflo proved that c 0 does not uniformly embed into Hilbert space.…”

Section: ) Uniform Embeddings and Smirnovmentioning

confidence: 99%

Metric cotype

2008

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We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe's theorem, settling a long standing open problem in the nonlinear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when L p coarsely or uniformly embeds into L q . We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to answer a question posed by Arora, Lovász, Newman, Rabani, Rabinovich and Vempala, and to obtain quantitative bounds in a metric Ramsey theorem due to Matoušek.

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Section: ) Uniform Embeddings and Smirnovmentioning

confidence: 99%

Metric cotype

2008

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“…Introduction. In 1974, Aharoni [1] proved that every separable metric space (M, d) is Lipschitz isomorphic to a subset of the Banach space c 0 . Thus, for some constant K, there is a map f : M → c 0 which satisfies the inequality d(x, y) ≤ f (x) − f (y) ≤ Kd(x, y), x, y ∈ M.…”

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confidence: 99%

Best constants for Lipschitz embeddings of metric spaces into c₀

Kalton

Lancien

2008

Fund. Math.

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Abstract. We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into c0 and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical p-spaces into c0 and give other applications. We prove that if a Banach space embeds almost isometrically into c0, then it embeds linearly almost isometrically into c0. We also study Lipschitz embeddings into c + 0 .

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“…I. Aharoni [1] proved that any separable metric space is Lipschitz equivalent to a subset of c 0 (in fact, to a subset of the positive cone of c 0 ). We concentrate our attention on the possibility of Lipschitz or uniform embeddings of spaces of continuous functions on a compact space into the space c 0 (Γ ) for sufficiently large Γ , i.e., equivalently, to C(α(Γ )), where α(L) denotes the Aleksandrov one-point compactification for any locally compact space L, and Γ is endowed with the discrete topology.…”

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confidence: 99%

C(K) spaces which cannot be uniformly embedded into c₀(Γ )

2006

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Abstract. We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c 0 (Γ ) for any set Γ . The first one is [0, ω 1 ] and hence it has the smallest possible cardinality, the other one has the smallest possible height ω 0 + 1.Foreword. We present two results of Jan Pelant. He presented them at seminars in the Mathematical Institute of Czech Academy of Sciences during the last two years. The example described in Theorem 4.1 below is quite recent. Unfortunately, Jan Pelant died before the results were prepared for publication. We decided to reconstruct them using his hand-written notes.1. Introduction. We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c 0 (Γ ) for any set Γ . This contributes to the study of nonlinear embeddings of (real) Banach spaces into other ones. This investigation is related to the study of the question which topological (or metric) properties enable one to reconstruct the linear structure of a Banach space.The well known Mazur-Ulam theorem says that the existence of an isometry of two Banach spaces implies their linear isometry. On the other hand, the result of H. Toruńczyk [17] (of M. I. Kadec [10] for separable spaces) shows that homeomorphism of two infinite-dimensional Banach spaces gives no information about their linear structure. This is the reason why uniform

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Every separable metric space is Lipschitz equivalent to a subset ofc 0 +

Cited by 74 publications

References 7 publications

Metric cotype

Metric cotype

Best constants for Lipschitz embeddings of metric spaces into c₀

C(K) spaces which cannot be uniformly embedded into c₀(Γ )

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Every separable metric space is Lipschitz equivalent to a subset ofc 0 +

Cited by 74 publications

References 7 publications

Metric cotype

Metric cotype

Best constants for Lipschitz embeddings of metric spaces into c0

C(K) spaces which cannot be uniformly embedded into c0(Γ )

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Product

Resources

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Best constants for Lipschitz embeddings of metric spaces into c₀

C(K) spaces which cannot be uniformly embedded into c₀(Γ )