The strong Brunn–Minkowski inequality and its equivalence with the $\mathsf{C}\mathsf{D}$ 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.