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Nearly Round Spheres Look Convex
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 1, February 2012
- pp. 109-139
- 10.1353/ajm.2012.0000
- Article
- Additional Information
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.