Nearly Round Spheres Look Convex

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Hölder Continuity and Injectivity of Optimal Maps (2011, background): “…‡Department of Mathematics, University of British Columbia, Vancouver BC Canada yhkim@math.ubc.ca and School of Mathematics, Institute for Advanced Study at Princeton NJ USA §Department of Mathematics, University of Toronto, Toronto Ontario Canada M5S 2E4 mccann@math.toronto.edu Contents”
Partial regularity for optimal transport maps (2012, background): “Let us also mention that, up to now, the MTW condition is known to be satisfied only for very special classes of Riemannian manifolds, such as spheres, their products, their quotients and submersions, and their perturbations [32, 22, 10, 29, 25, 20, 11], and for instance it is known to fail on suffi”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “Thus −(vc, v) is a better candidate for a maximizer than −(u, v).”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “Some examples of manifolds satisfying the MTW condition have been found in [55, 57, 73, 74, 84, 85]: • R and T satisfy MTW(0).”
On the Ma–Trudinger–Wang curvature on surfaces (2010, background): “Examples of regular manifolds appear in [10, 11, 23, 24, 19].”