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.

中文翻译：

---

次黎曼结构不满足黎曼Brunn-Minkowski不等式

我们证明，严格的次黎曼结构不能满足曲率尺寸和最优输运的黎曼理论中的Brunn-Minkowski不等式。我们的证明依赖于与Heisenberg组相同的方法，以及Agrachev，Barilari和Rizzi对新的黎曼子结构的充足法线大地测量学以及与它们相连的大地测量尺度的新研究。