Geodesic
Stability for the Brunn-Minkowski and Riesz Rearrangement Inequalities, with Applications to Gaussian Concentration and Finite Range Non-local Isoperimetry
Canadian journal of mathematics, Tome 69 (2017) no. 5, pp. 1036-1063

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DOI

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.
DOI : 10.4153/CJM-2016-026-9
Mots-clés : 49N99, Brunn-Minkowski inequality, Riesz rearrangement, Gaussian Concentration, Gates-Penrose-Lebowitz energy 
Carlen, Eric; Maggi, Francesco. Stability for the Brunn-Minkowski and Riesz Rearrangement Inequalities, with Applications to Gaussian Concentration and Finite Range Non-local Isoperimetry. Canadian journal of mathematics, Tome 69 (2017) no. 5, pp. 1036-1063. doi: 10.4153/CJM-2016-026-9
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     author = {Carlen, Eric and Maggi, Francesco},
     title = {Stability for the {Brunn-Minkowski} and {Riesz} {Rearrangement} {Inequalities,} with {Applications} to {Gaussian} {Concentration} and {Finite} {Range} {Non-local} {Isoperimetry}},
     journal = {Canadian journal of mathematics},
     pages = {1036--1063},
     year = {2017},
     volume = {69},
     number = {5},
     doi = {10.4153/CJM-2016-026-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2016-026-9/}
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