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

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.