# Isomorphic uniform convexity in metric spaces

# Isomorphic uniform convexity in metric spaces

In 1985 Bourgain gave a discrete metric characterization of when a normed space admits an equivalent uniformly convex norm: this happens if and only if the space does not contain arbitrarily large complete binary trees with low distortion. Bourgain's theorem was the first successful step in a research program that was inspired by a theorem of Ribe from 1976, which suggested that certain linear properties of normed spaces are actually metric properties in disguise. In this talk we will revisit Bourgain's solution in a quantitative way: we will obtain a metric characterization of the degree to which a norm is uniformly convex. It turns out that this question exhibits unexpected subtleties and surprising differences from the linear theory of normed spaces and from the theory of metric type and cotype. Nevertheless, we will be able to show that a normed space has an equivalent norm with modulus of uniform convexity of power type p if and only if a concrete metric inequality called Markov p-Convexity is satisfied. This inequality controls the rate at which certain Markov chains diverge in a metric space, and it is useful for problems in metric geometry which do not necessarily involve linear spaces. We will discuss some of these applications, as well as some unexpected counterexamples. Joint work with Manor Mendel.