Inventiones mathematicae

, Volume 182, Issue 1, pp 167–211 | Cite as

A mass transportation approach to quantitative isoperimetric inequalities

  • A. Figalli
  • F. Maggi
  • A. Pratelli


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.


Sobolev Inequality Isoperimetric Inequality Asymmetry Index Trace Inequality Coarea Formula 
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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Texas at AustinAustinUSA
  2. 2.Dipartimento di Matematica “U. Dini”Università di FirenzeFirenzeItaly
  3. 3.Dipartimento di MatematicaUniversità di PaviaPaviaItaly

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