Markov Type constants, flat tori and Wasserstein spaces

TL;DR

This paper investigates Markov type constants in metric spaces, establishing inequalities under coverings, quotients, and Wasserstein space constructions, with applications to flat manifolds and Wasserstein space embeddings.

Contribution

It introduces new bounds for Markov type constants under coverings, quotients, and Wasserstein space formations, solving open questions and confirming conjectures.

Findings

All compact flat manifolds have Markov type 2 with constant 1.

Circle with intrinsic metric has Markov type 2 with constant 1.

New upper bounds for Wasserstein space Markov type constants, confirming conjectures.

Abstract

Let $M_p(X,T)$ denote the Markov type $p$ constant at time $T$ of a metric space $X$, where $p \ge 1$. We show that $M_p(Y,T) \le M_p(X,T)$ in each of the following cases: (a)$X$ and $Y$ are geodesic spaces and $Y$ is covered by $X$ via a finite-sheeted locally isometric covering, (b)$Y$ is the quotient of $X$ by a finite group of isometries, (c) $Y$ is the $L^p$-Wasserstein space over $X$. As an application of (a) we show that all compact flat manifolds have Markov type $2$ with constant $1$. In particular the circle with its intrinsic metric has Markov type $2$ with constant $1$. This answers the question raised by S.-I. Ohta and M. Pichot. Parts (b) and (c) imply new upper bounds for Markov type constants of the $L^p$-Wasserstein space over $\mathbb R^d$. These bounds were conjectured by A. Andoni, A. Naor and O. Neiman. They imply certain restrictions on bi-Lipschitz embeddability of snowflakes into such Wasserstein spaces.