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A new metric invariant for Banach spaces

F. Baudier, N. J. Kalton, G. Lancien (2010)

Studia Mathematica

We show that if the Szlenk index of a Banach space X is larger than the first infinite ordinal ω or if the Szlenk index of its dual is larger than ω, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into X. We show that the converse is true when X is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms.

A note on Lipschitz isomorphisms in Hilbert spaces

Dean Ives (2010)

Commentationes Mathematicae Universitatis Carolinae

We show that the following well-known open problems on existence of Lipschitz isomorphisms between subsets of Hilbert spaces are equivalent: Are balls isomorphic to spheres? Is the whole space isomorphic to the half space?

Lipschitz and uniform embeddings into

N. J. Kalton (2011)

Fundamenta Mathematicae

We show that there is no uniformly continuous selection of the quotient map Q : / c relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from B X * * onto B X .

Markov convexity and local rigidity of distorted metrics

Manor Mendel, Assaf Naor (2013)

Journal of the European Mathematical Society

It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type p if and only if it is Markov p -convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity of metric spaces, showing in particular that tree metrics fail to have the dichotomy property.

Perturbations of isometries between Banach spaces

Rafał Górak (2011)

Studia Mathematica

We prove a very general theorem concerning the estimation of the expression ||T((a+b)/2) - (Ta+Tb)/2|| for different kinds of maps T satisfying some general perturbed isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone...

Sphere equivalence, Property H, and Banach expanders

Qingjin Cheng (2016)

Studia Mathematica

We study the uniform classification of the unit spheres of general Banach sequence spaces. In particular, we obtain some interesting applications involving Property H introduced by Kasparov and Yu, and Banach expanders.

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