Nearly Round Spheres Look Convex

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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