# Generalized barycenters and variance maximization on metric spaces

arXiv: https://arxiv.org/abs/2006.02984 · HTML: https://mlsanity.com/paper/2006.02984/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 May 2020.

## Built on by (most cited first, 1 of 1)

- [Uncertainty Quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ball](https://mlsanity.com/paper/2108.10517/) (2021)

## Abstract (arXiv)

We show that the variance of a probability measure μ 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) × 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.