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.
A mass transportation approach to quantitative isoperimetric inequalities / A. Figalli; F. Maggi; A. Pratelli. - In: INVENTIONES MATHEMATICAE. - ISSN 0020-9910. - STAMPA. - 182:(2010), pp. 167-211. [10.1007/s00222-010-0261-z]
A mass transportation approach to quantitative isoperimetric inequalities
MAGGI, FRANCESCO;
2010
Abstract
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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