The Brunn--Minkowski inequality implies the CD condition in weighted Riemannian manifolds

TL;DR

This paper proves that in weighted Riemannian manifolds, the Brunn--Minkowski inequality is equivalent to the curvature dimension condition, enabling characterization without optimal transport or differential structures.

Contribution

It establishes the equivalence between BM(K,N) and CD(K,N) in weighted Riemannian manifolds, removing the need for optimal transport or differential tools.

Findings

BM(K,N) is equivalent to CD(K,N) in weighted Riemannian manifolds

Characterization of curvature dimension condition without optimal transport

Extension of the Brunn--Minkowski inequality implications

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.