The dimensional Brunn–Minkowski inequality in Gauss space

Let γn be the standard Gaussian measure on Rn. We prove that for every symmetric convex sets K,L in Rn and every λ∈(0,1),γn(λK+(1−λ)L)1n⩾λγn(K)1n+(1−λ)γn(L)1n, thus settling a problem raised by Gardner and Zvavitch (2010). This is the Gaussian analogue of the classical Brunn–Minkowski inequality for...

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Veröffentlicht in:	Journal of functional analysis 2021-03, Vol.280 (6), p.108914, Article 108914
Hauptverfasser:	Eskenazis, Alexandros, Moschidis, Georgios
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Sprache:	eng
Schlagworte:	Brunn–Minkowski inequality Gardner–Zvavitch problem Gaussian measure Mathematics Symmetric convex sets
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creator	Eskenazis, Alexandros Moschidis, Georgios
description	Let γn be the standard Gaussian measure on Rn. We prove that for every symmetric convex sets K,L in Rn and every λ∈(0,1),γn(λK+(1−λ)L)1n⩾λγn(K)1n+(1−λ)γn(L)1n, thus settling a problem raised by Gardner and Zvavitch (2010). This is the Gaussian analogue of the classical Brunn–Minkowski inequality for the Lebesgue measure. We also show that, for a fixed λ∈(0,1), equality is attained if and only if K=L.
doi_str_mv	10.1016/j.jfa.2020.108914
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subjects	Brunn–Minkowski inequality Gardner–Zvavitch problem Gaussian measure Mathematics Symmetric convex sets
title	The dimensional Brunn–Minkowski inequality in Gauss space
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