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