A mass transportation approach to quantitative isoperimetric inequalities
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- Figalli, A., Maggi, F. & Pratelli, A. Invent. math. (2010) 182: 167. doi:10.1007/s00222-010-0261-z
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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.