On supporting hyperplanes to convex bodies
  Given a convex set and an interior point close to the boundary, we prove the existence of a supporting hyperplane whose distance to the point is controlled, in a dimensionally quantified way, by the thickness of the convex set in the orthogonal direction. This result has important applications in the regularity theory for Monge-Amp\`ere type equations arising in optimal transportation. 
Alessio Figalli, Young-Heon Kim and Robert J. McCann
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