We discuss the equivalence between the curvature dimension condition CD(K,N), in the sense of Lott--Sturm--Villani, and the validity of the generalised Brunn--Minkowski inequality BM(K,N). As a first step, we prove such equivalence in the setting of weighted Riemannian manifolds, where the CD(K,N) condition admits a nice, differential riformulation in terms of lower bound on a modified Ricci tensor. In the setting of essentially non-branching metric measure spaces, the equivalence is still an open problem. In this talk, we present a preliminary result in this direction, showing that at this level of generality, the CD(K,N) condition is a equivalent to a newly introduced notion that we call strong Brunn--Minkowski inequality SBM(K,N), which is a reinforcement of the generalized Brunn--Minkowski inequality BM(K,N).