The strong Brunn--Minkowski inequality and its equivalence with the CD condition

TL;DR

This paper establishes the equivalence between the curvature-dimension condition and a new strong Brunn--Minkowski inequality in non-branching metric measure spaces, advancing understanding of geometric analysis in these spaces.

Contribution

It introduces the strong Brunn--Minkowski inequality and proves its equivalence with the CD(K,N) condition in non-branching metric measure spaces.

Findings

Equivalence between CD(K,N) and SBM(K,N) in non-branching spaces

Reinforcement of the generalized Brunn--Minkowski inequality

Progress towards full equivalence with BM(K,N) in weighted Riemannian manifolds

Abstract

In the setting of essentially non-branching metric measure spaces, we prove the equivalence between the curvature dimension condition CD(K,N), in the sense of Lott--Sturm--Villani, and a newly introduced notion that we call strong Brunn--Minkowski inequality SBM(K,N). This condition is a reinforcement of the generalized Brunn--Minkowski inequality BM(K,N), which is known to hold in CD(K,N) spaces. Our result is a first step towards providing a full equivalence between the CD(K,N) condition and the validity of BM(K,N), which we have recently proved in the framework of weighted Riemannian manifolds.