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On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities


Authors: Galyna Livshyts, Arnaud Marsiglietti, Piotr Nayar and Artem Zvavitch
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 52A40; Secondary 60G15
DOI: https://doi.org/10.1090/tran/6928
Published electronically: April 11, 2017
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Abstract: In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality

$\displaystyle \mu (\lambda A + (1-\lambda )B)^{1/n} \geq \lambda \mu (A)^{1/n} + (1-\lambda )\mu (B)^{1/n} $

holds true for an unconditional product measure $ \mu $ with non-increasing density and a pair of unconditional convex bodies $ A,B \subset \mathbb{R}^n$. We also show that the above inequality is true for any unconditional $ \log $-concave measure $ \mu $ and unconditional convex bodies $ A,B \subset \mathbb{R}^n$. Finally, we prove that the inequality is true for a symmetric $ \log $-concave measure $ \mu $ and a pair of symmetric convex sets $ A,B \subset \mathbb{R}^2$, which, in particular, settles the two-dimensional case of the conjecture for Gaussian measure proposed by Gardner and Zvavitch in 2010.

In addition, we note that in the cases when the above inequality is true, one can deduce from it the $ 1/n$-concavity of the parallel volume $ t \mapsto \mu (A+tB)$, Brunn's type theorem and certain analogues of Minkowski's first inequality.


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Additional Information

Galyna Livshyts
Affiliation: 228 Skiles building, School of sciences, Georgia Institute of Technology, 686 Cherry Street NW, Atlanta, Georgia 30332
Email: glivshyts6@math.gatech.edu

Arnaud Marsiglietti
Affiliation: Institute for Mathematics and Its Applications, University of Minnesota, 207 Church Street SE, 434 Lind Hall, Minneapolis, Minnesota 55455
Email: arnaud.marsiglietti@ima.umn.edu

Piotr Nayar
Affiliation: Wharton Department of Statistics, University of Pennsylvania, 432-1 Jon M. Huntsman Hall, 3730 Walnut Street, Philadelphia, Pennsylvania 19104
Email: nayar@mimuw.edu.pl

Artem Zvavitch
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: zvavitch@math.kent.edu

DOI: https://doi.org/10.1090/tran/6928
Keywords: Convex body, Gaussian measure, Brunn-Minkowski inequality, Minkowski first inequality, S-inequality, Brunn's theorem, Gaussian isoperimetry, log-Brunn-Minkowski inequality
Received by editor(s): July 9, 2015
Received by editor(s) in revised form: February 7, 2016, and February 25, 2016
Published electronically: April 11, 2017
Additional Notes: The first author was supported in part by the U.S. National Science Foundation Grant DMS-1101636
The second author was supported in part by the Institute for Mathematics and Its Applications with funds provided by the National Science Foundation
The third author was supported in part by NCN grant DEC-2012/05/B/ST1/00412
The fourth author was supported in part by the U.S. National Science Foundation Grant DMS-1101636 and the Simons Foundation
Article copyright: © Copyright 2017 American Mathematical Society