Generalized Variance Inequalities for barycenters in CAT(0) and CAT(1) spaces
Abstract
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 radii less than π /2 in complete CAT (1) spaces and to exponents p ≥ 2 in the CAT (0) setting. Building on a result of Eskenazis, Mendel, and Naor, we establish sharp metric cotype for all p ≥ 2 in 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 CAT(0) spaces and balls of radii less than π /2 in complete CAT (1) spaces.
Fields of Study & Subjects
Advanced Differential Geometry Research
Geometric Analysis and Curvature Flows
Advanced Banach Space Theory
Publication details
- Journal / source
- Analysis and Geometry in Metric Spaces
- Publisher
- Walter de Gruyter GmbH
- Volume
- 14
- Issue
- 1
- Publication date
- 1 January 2026
- Language
- EN