Sub-Riemannian interpolation inequalities

We prove that ideal sub-Riemannian manifolds (i.e., admitting no non-trivial abnormal minimizers) support interpolation inequalities for optimal transport. A key role is played by sub-Riemannian Jacobi fields and distortion coefficients, whose properties are remarkably different with respect to the Riemannian case. As a byproduct, we characterize the cut locus as the set of points where the squared sub-Riemannian distance fails to be semiconvex, answering to a question raised by Figalli and Rifford (Geom Funct Anal 20(1):124–159, 2010). As an application, we deduce sharp and intrinsic Borell–Brascamp–Lieb and geodesic Brunn–Minkowski inequalities in the aforementioned setting. For the case of the Heisenberg group, we recover in an intrinsic way the results recently obtained by Balogh et al. (Calc Var Part Differ Equ 57(2):61, 2018), and we extend them to the class of generalized H-type Carnot groups. Our results do not require the distribution to have constant rank, yielding for the particular case of the Grushin plane a sharp measure contraction property and a sharp Brunn–Minkowski inequality.