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

Abstract

The curvature dimension condition CD(K,N), pioneered by Sturm and Lott–Villani in Sturm (2006a); Sturm (2006b); Lott and Villani (2009), 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 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.