Nearly round spheres look convex
type: Item
instance of: scholarly article
zbMATH Open document ID: 1241.53031
author: Alessio Figalli
author: Ludovic Rifford
author: Cédric Villani
published in: American Journal of Mathematics
publication date: 2012-03-29T00:00:00Z
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53B21
zbMATH DE Number: 6019023
zbMATH Keywords: Riemannian metric
zbMATH Keywords: round sphere
zbMATH Keywords: injectivity domain
zbMATH Keywords: nonfocal domain
zbMATH Keywords: Ma-Trudinger-Wang tensor
MaRDI profile type: Publication
full work available at URL: ajm.2012.0000
OpenAlex ID: W2081168151
DOI: 10.1353/AJM.2012.0000
Recommended article: Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2
Recommended article: Regularity of optimal transport maps on multiple products of spheres
Recommended article: Boundary $C^{1,\alpha}$ regularity of an optimal transport problem with cost close to $-x\cdot y$
Recommended article: Stability of a 4th-order curvature condition arising in optimal transport theory
Recommended article: Q3076577
Recommended article: On the local geometry of maps with c-convex potentials
Recommended article: On the injectivity and nonfocal domains of the ellipsoid of revolution
Recommended article: Regularity of optimal transport maps on locally nearly spherical manifolds
Recommended article: On the regularity of optimal transportation potentials on round spheres
Recommended article: Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna
title: Nearly round spheres look convex
review text: 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.
reviewed by: Harold Parks
description: scientific article; zbMATH DE number 6019023