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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.
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Nicolas Juillet, Sub-Riemannian structures do not satisfy Riemannian Brunn–Minkowski inequalities. Rev. Mat. Iberoam. 37 (2021), no. 1, pp. 177–188DOI 10.4171/RMI/1205