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

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 $p\ge 2$ in the $\mathrm{CAT}(0)$ setting. Building on a result of Eskenazis, Mendel, and Naor, we establish sharp metric cotype for all $p\ge 2$ in $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ 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 $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ space and balls of sufficiently small radius in complete CAT(1) spaces.