639 research outputs found
Metric Cotype
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 L_q into L_p, when q > max{2,p}. Finally we use metric cotype
to obtain a new type of isoperimetric inequality on the discrete
torus
Metric Cotype
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, Lovasz, Newman, Rabani, Rabinovich and Vempala, and to obtain
quantitative bounds in a metric Ramsey theorem due to Matousek.Comment: 46 pages. Fixes the layou
Spaces of small metric cotype
Naor and Mendel's metric cotype extends the notion of the Rademacher cotype
of a Banach space to all metric spaces. Every Banach space has metric cotype at
least 2. We show that any metric space that is bi-Lipschitz equivalent to an
ultrametric space has infinimal metric cotype 1. We discuss the invariance of
metric cotype inequalities under snowflaking mappings and Gromov-Hausdorff
limits, and use these facts to establish a partial converse of the main result.Comment: 14 pages, the needed isometric inequality now derived from the
literature, rather than proved by hand; other minor typos and errors fixe
Improved bounds in the metric cotype inequality for Banach spaces
It is shown that if (X, ||.||_X) is a Banach space with Rademacher cotype q
then for every integer n there exists an even integer m< n^{1+1/q}\sum_{j=1}^n \Avg_x [ ||f(x+ (m/2) e_j)-f(x)
||_X^q ] < C m^q \Avg_{\e,x} [ ||f(x+\e)-f(x) ||_X^q ]$, where the expectations
are with respect to uniformly chosen x\in Z_m^n and \e\in \{-1,0,1\}^n, and all
the implied constants may depend only on q and the Rademacher cotype q constant
of X. This improves the bound of m< n^{2+\frac{1}{q}} from [Mendel, Naor 2008].
The proof of the above inequality is based on a "smoothing and approximation"
procedure which simplifies the proof of the metric characterization of
Rademacher cotype of [Mendel, Naor 2008]. We also show that any such "smoothing
and approximation" approach to metric cotype inequalities must require m>
n^{(1/2)+(1/q)}.Comment: 27 pages, 1 figure. Fixes a slight error in the proof of Lemma 4.3 in
the arXiv v2 and the published pape
Embeddings of locally finite metric spaces into Banach spaces
We show that if X is a Banach space without cotype, then every locally finite
metric space embeds metrically into X.Comment: 6 pages, to appear in Proceedings of the AM
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