SubRiemanniann structures do not satisify Riemannian Brunn--Minkowski inequalities

TL;DR

This paper demonstrates that subRiemannian structures cannot satisfy Riemannian Brunn--Minkowski inequalities, highlighting fundamental differences between these geometric frameworks and impacting the understanding of curvature and optimal transport in subRiemannian geometry.

Contribution

It proves the non-existence of Riemannian Brunn--Minkowski inequalities in strictly subRiemannian structures using novel methods and recent insights into geodesic properties.

Findings

SubRiemannian structures do not satisfy Riemannian Brunn--Minkowski inequalities.

The proof extends techniques from the Heisenberg group to general subRiemannian cases.

New investigations into geodesic dimensions support the main result.

Abstract

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.