The dimensional Brunn–Minkowski inequality in Gauss space

By Alexandros Eskenazis, Georgios Moschidis. Journal of Functional Analysis, 2020

56 citations recorded by the source corpus.

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$. We prove that for every symmetric convex sets $K,L$ in $\mathbb{R}^n$ and every $\lambda\in(0,1)$, $$\gamma_n(\lambda K+(1-\lambda)L)^{\frac{1}{n}} \geq \lambda \gamma_n(K)^{\frac{1}{n}}+(1-\lambda)\gamma_n(L)^{\frac{1}{n}},$$ 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 $\lambda\in(0,1)$, equality is attained if and only if $K=L$.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/3d874ae7ae09ff32f9efc529cf6f509d95812477

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Citations in the corpus

• The Minkowski problem in Gaussian probability space (2020, background): “Of particular interest is the dimensional Brunn-Minkowski inequality (for o-symmetric convex bodies) conjectured by Gardner-Zvavitch [25], with important contribution by Kolesnikov-Livshyts [45] followed by a recent confirmation by Eskenazis-Moschidis [21].”