Generalized barycenters and variance maximization on metric spaces

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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.