Barycenter curvature-dimension condition for extended metric measure spaces

TL;DR

This paper introduces the Barycenter-Curvature Dimension (BCD) condition for extended metric measure spaces, providing a new framework based on Wasserstein barycenters to analyze curvature and dimension properties.

Contribution

It proposes the BCD condition, a novel curvature-dimension criterion for extended metric measure spaces, from the perspective of Wasserstein barycenters.

Findings

Defines the BCD condition for extended metric measure spaces

Establishes foundational properties of BCD

Provides insights into curvature-dimension analysis using Wasserstein barycenters

Abstract

In this survey, we introduce a new curvature-dimension condition for extended metric measure spaces, called Barycenter-Curvature Dimension condition BCD, from the perspective of Wasserstein barycenter.