Published Paper

Inserted: 28 sep 2022
Last Updated: 26 feb 2024

Journal: Nonlinear Analysis
Year: 2024
Doi: https://doi.org/10.1016/j.na.2024.113502

Abstract:

The curvature dimension condition CD(K,N), pioneered by Sturm and Lott--Villani, is a synthetic notion of having curvature bounded below and dimension bounded above, in the non-smooth setting. This condition implies a suitable generalization of the Brunn--Minkowski inequality, denoted by BM(K,N). In this paper, we address the converse implication in the setting of weighted Riemannian manifolds, proving that BM(K,N) is in fact equivalent to CD(K,N). Our result allows to characterize the curvature dimension condition without using neither the optimal transport nor the differential structure of the manifold.