Inventiones mathematicae

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

A mass transportation approach to quantitative isoperimetric inequalities

  • A. FigalliEmail author
  • 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© 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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