Calculus of Variations and Geometric Measure Theory

M. Magnabosco - L. Portinale - T. Rossi

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

created by portinale on 28 Sep 2022
modified on 26 Feb 2024


Published Paper

Inserted: 28 sep 2022
Last Updated: 26 feb 2024

Journal: Nonlinear Analysis
Year: 2024

ArXiv: 2209.13424 PDF


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.