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

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- by Galyna Livshyts, Arnaud Marsiglietti, Piotr Nayar and Artem Zvavitch PDF
- Trans. Amer. Math. Soc.
**369**(2017), 8725-8742 Request permission

## 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 \[ \mu (\lambda A + (1-\lambda )B)^{1/n} \geq \lambda \mu (A)^{1/n} + (1-\lambda )\mu (B)^{1/n} \] 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
- MR Author ID: 1015863
- 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
- MR Author ID: 1063405
- 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
- MR Author ID: 890939
- Email: nayar@mimuw.edu.pl
**Artem Zvavitch**- Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
- MR Author ID: 671170
- Email: zvavitch@math.kent.edu
- 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 - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 8725-8742 - MSC (2010): Primary 52A40; Secondary 60G15
- DOI: https://doi.org/10.1090/tran/6928
- MathSciNet review: 3710641