Generalized barycenters and variance maximization on metric spaces

TL;DR

This paper establishes bounds on the variance of probability measures in metric spaces using geometric embeddings, extending classical results like Jung's theorem to CAT(k) spaces through minimax theory.

Contribution

It introduces a novel approach linking variance bounds to circumradii in metric spaces via Wasserstein embeddings, extending results to general moments and non-Euclidean spaces.

Findings

Variance bounded by circumradius squared in metric spaces

Extension of Jung's theorem to CAT(k) spaces

Application of minimax theory to probability measure embeddings

Abstract

We show that the variance of a probability measure $\mu$ on a compact subset $X$ of a complete metric space $M$ is bounded by the square of the circumradius $R$ of the canonical embedding of $X$ into the space $P(M)$ of probability measures on $M$, equipped with the Wasserstein metric. When barycenters of measures on $X$ are unique (such as on CAT($0$) spaces), our approach shows that $R$ in fact coincides with the circumradius of $X$ and so this result extends a recent result of Lim-McCann from Euclidean space. Our approach involves bi-linear minimax theory on $P(X) \times P(M)$ and extends easily to the case when the variance is replaced by very general moments. As an application, we provide a simple proof of Jung's theorem on CAT($k$) spaces, a result originally due to Dekster and Lang-Schroeder.