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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:2016-11-08
 Printed: Oct 2017
  • Eric Carlen,
    Department of Mathematics, Rutgers University , 110 Frelinghuysen Road, Piscataway NJ 08854-8019, USA
  • Francesco Maggi,
    Department of Mathematics, University of Texas , Austin, TX 78712
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Abstract

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: Brunn-Minkowski inequality, Riesz rearrangement, Gaussian Concentration, Gates-Penrose-Lebowitz energy Brunn-Minkowski inequality, Riesz rearrangement, Gaussian Concentration, Gates-Penrose-Lebowitz energy
MSC Classifications: 49N99 show english descriptions None of the above, but in this section 49N99 - None of the above, but in this section
 

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