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

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 [Sturm, On the geometry of metric measure spaces. I, Acta Math. 196(1) (2006) 65-131; On the geometry of metric measure spaces. II, Acta Math. 196(1) (2006) 133-177; Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169(3) (2009) 903-991], 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 toward providing a full equivalence between the CD(K,N) condition and the validity of BM(K,N), which has been recently proved in [M. Magnabosco, L. Portinale and T. Rossi, The Brunn-Minkowski inequality implies the CD condition in weighted Riemannian manifolds, Nonlinear Anal. 242 (2024) 113502] in the framework of weighted Riemannian manifolds.