Methods and Applications of Analysis

Volume 20 (2013)

Number 3

On supporting hyperplanes to convex bodies

Pages: 261 – 272



Alessio Figalli (Department of Mathematics, University of Texas, Austin Tx., U.S.A.)

Young-Heon Kim (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)

Robert J. McCann (Department of Mathematics, University of Toronto, Ontario Canada)


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-Ampère type equations arising in optimal transportation.


convex bodies, supporting hyperplanes

2010 Mathematics Subject Classification

35J96, 52A20, 52A40

Published 12 November 2013