The strong Brunn–Minkowski inequaliity 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 Formula: see text, 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 Formula: see text. This condition is a reinforcement of the generalized Brunn–Minkowski inequality Formula: see text, which is known to hold in Formula: see text spaces. Our result is a first step toward providing a full equivalence between the Formula: see text condition and the validity of Formula: see text, 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.