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Stability for the Brunn-Minkowski and Riesz Rearrangement Inequalities, with Applications to Gaussian Concentration and Finite Range Non-local Isoperimetry

Published online by Cambridge University Press:  20 November 2018

Eric Carlen  and
Francesco Maggi
Eric Carlen
Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway NJ 08854-8019, USA e-mail: carlen@math.rutgers.edu
Francesco Maggi
Affiliation:
Department of Mathematics, University of Texas, AustinTX 78712 e-mail: maggi@math.utexas.edu
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Abstract

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We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is then used to obtain a robust improvement of the Riesz rearrangement inequality under certain natural conditions. These conditions are compatible with the applications to a finite-range nonlocal isoperimetric problem arising in statistical mechanics.

Keywords

49N99Brunn-Minkowski inequalityRiesz rearrangementGaussian ConcentrationGates-Penrose-Lebowitz energy
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 5 , 01 October 2017 , pp. 1036 - 1063
Copyright
Copyright © Canadian Mathematical Society 2017

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