Inventiones mathematicae

, Volume 182, Issue 1, pp 167–211

A mass transportation approach to quantitative isoperimetric inequalities


    • Department of MathematicsThe University of Texas at Austin
  • F. Maggi
    • Dipartimento di Matematica “U. Dini”Università di Firenze
  • A. Pratelli
    • Dipartimento di MatematicaUniversità di Pavia

DOI: 10.1007/s00222-010-0261-z

Cite this article as:
Figalli, A., Maggi, F. & Pratelli, A. Invent. math. (2010) 182: 167. doi:10.1007/s00222-010-0261-z


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.

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© Springer-Verlag 2010