Generalized Variance Inequalities for Barycenters in CAT(0) and CAT(1) Spaces

Abstract

We prove generalized versions of the Variance Inequality known for barycenters in CAT(0) spaces, inspired by an analogous result for $p$-uniformly convex Banach spaces. Our generalizations apply to balls of sufficiently small radius in complete CAT(1) spaces and to exponents $p \geq 2$ in the $\operatorname{CAT}(0)$ setting. Building on a result of Eskenazis, Mendel, and Naor, we establish sharp metric cotype for all $p \geq 2$ in $\mathrm{CAT}(0)$ spaces, extending the previously known case $p=2$. In addition, based on their work, we derive martingale inequalities for nonlinear martingales taking values in complete $\mathrm{CAT}(0)$ space and balls of sufficiently small radius in complete CAT(1) spaces.