# Sub-Riemannian interpolation inequalities

## TL;DR

This paper establishes interpolation inequalities for ideal sub-Riemannian manifolds, characterizing the cut locus and deriving sharp geometric inequalities, extending known results to broader classes like the Grushin plane.

## Contribution

It introduces new interpolation inequalities in sub-Riemannian geometry, characterizes the cut locus, and extends geometric inequalities to generalized H-type Carnot groups.

## Key findings

- Characterization of the cut locus as points where squared distance is not semiconvex.
- Derivation of sharp Borell-Brascamp-Lieb inequalities in sub-Riemannian settings.
- Extension of measure contraction and Brunn-Minkowski inequalities to Grushin plane.

## Abstract

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 in [Geom. Funct. Anal. (2010) 20: 124]. 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, Krist\'aly and Sipos in [Calc. Var. PDE (2018) 57: 61], 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.