Nearly round spheres look convex

Abstract : We prove that a Riemannian manifold $(M,g)$, close enough to the round sphere in the $C^4$ topology, has uniformly convex injectivity domains---so $M$ appears uniformly convex in any exponential chart. The proof is based on the Ma-Trudinger-Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport.
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Contributor : Aurélie Reymond <>
Submitted on : Monday, April 7, 2014 - 2:13:19 PM
Last modification on : Wednesday, December 12, 2018 - 3:32:50 PM

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Alessio Figalli, Ludovic Rifford, Cédric Villani. Nearly round spheres look convex. American Journal of Mathematics, Johns Hopkins University Press, 2012, 134 (1), pp.109-139. ⟨10.1353/ajm.2012.0000⟩. ⟨hal-00974756⟩



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