Sub-Riemannian structures do not satisfy Riemannian Brunn–Minkowski inequalities

Abstract

We prove that no Brunn–Minkowski inequality from the Riemannian theories of curvature-dimension and optimal transportation can be satisfied by a strictly sub-Riemannian structure. Our proof relies on the same method as for the Heisenberg group together with new investigations by Agrachev, Barilari and Rizzi on ample normal geodesics of sub-Riemannian structures and the geodesic dimension attached to them.