Inserted: 12 nov 2007
Last Updated: 16 feb 2015
Journal: Invent. Math.
A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov's proof of the isoperimetric inequality and the Brenier-McCann Theorem. A sharp quantitative version of the Brunn-Minkowski inequality for convex sets is proved as a corollary.
Keywords: Sets of finite perimeter, Mass transportation, isoperimetric inequality, wulff shape