Inventiones mathematicae

, Volume 182, Issue 1, pp 167-211

First online:

A mass transportation approach to quantitative isoperimetric inequalities

  • A. FigalliAffiliated withDepartment of Mathematics, The University of Texas at Austin Email author 
  • , F. MaggiAffiliated withDipartimento di Matematica “U. Dini”, Università di Firenze
  • , A. PratelliAffiliated withDipartimento di Matematica, Università di Pavia

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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.