Markov convexity and local rigidity of distorted metrics

TL;DR

This paper establishes a characterization of Banach spaces with power-type uniform convexity norms through Markov p-convexity, and provides counterexamples to certain isomorphic convexity questions in metric spaces.

Contribution

It introduces a new equivalence between Markov p-convexity and power-type uniform convexity in Banach spaces, and constructs counterexamples to natural conjectures in metric space theory.

Findings

Banach spaces with power-type p convexity are characterized by Markov p-convexity.

Counterexamples show tree metrics do not have the dichotomy property.

The paper links geometric properties of Banach spaces with probabilistic Markov properties.

Abstract

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.