Sub-Riemannian structures can be described as limits of Riemannian ones with $\mathrm{Ric}(g_n) \to -\infty$ and they represent, in a certain sense, the most singular case among the three great classes of geometries (Riemannian, Finlser, and sub-Riemannian ones). In this talk, we discuss how, under generic assumptions, these structures support interpolation inequalities \`a la Cordero\--Erasquin\--McCann\--Schmuckenschl\"ager. As a byproduct, we characterize the sub-Riemannian cut locus as the set of points where the squared sub-Riemannian distance fails to be semiconvex. Specifying our results to the case of the Heisenberg groups, we recover in an intrinsic way the inequalities recently obtained by Balogh, Krist\'aly and Sipos. As a further application, we obtain new and sharp results on the measure contraction properties of the standard Grushin structure. The techniques are based on optimal transport and sub-Riemannian Jacobi fields. Joint work with Davide Barilari.

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