A mass transportation approach to quantitative isoperimetric inequalities

• DOI: 10.1007/s00222-010-0261-z
• Publication Date: 2010-05-31T07:21:37Z
• Authors: A. Figalli; F. Maggi; A. Pratelli
• Keywords: Mass transportation

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.