Nearly round spheres look convex
In an earlier paper [Commun. Pure Appl. Math. 62, No. 12, 1670--1706 (2009; Zbl 1175.49040)], the first two authors studied the regularity of the optimal transport map between two probability measures in the case in which the cost is given by the squared Riemannian distance. In the same paper, as a surprising corollary, they deduced that for a perturbation of the round \(2\)-sphere all the injectivity domains are strictly convex. In the paper under review, the earlier result concerning perturbations of round \(2\)-spheres is extended to perturbations of round spheres of arbitrary dimension. More precisely, the main result of the paper is that for a Riemannian manifold \((M,g)\) where \(M\) is the \(n\)-sphere, if the metric \(g\) is sufficiently close in the \(C^4\)-norm to the round metric, then all injectivity domains in \((M,g)\) are uniformly convex. An important piece of the argument involves proving that a \(C^4\) perturbation of the round sphere satisfies an extended uniform Ma--Trudinger--Wang condition. The authors have sought to keep their proofs self-contained, and they have included cogent introductory material that puts their results in context.
scientific article; zbMATH DE number 6019023
scientific article