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A mass transportation approach to quantitative isoperimetric inequalities

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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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Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX, 78712, USA

    A. Figalli

  2. Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

    F. Maggi

  3. Dipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100, Pavia, Italy

    A. Pratelli

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  1. A. Figalli
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  2. F. Maggi
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  3. A. Pratelli
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Correspondence to A. Figalli.

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Figalli, A., Maggi, F. & Pratelli, A. A mass transportation approach to quantitative isoperimetric inequalities. Invent. math. 182, 167–211 (2010). https://doi.org/10.1007/s00222-010-0261-z

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