An extension of Milman's reverse Brunn-Minkowski inequality

Abstract

The classical Brunn-Minkowski inequality states that for $A_1,A_2\subset {\mathbb{R}}^n$ compact, $$ |A_1+A_2|^{1/n}\ge |A_1|^{1/n}+|A_2|^{1/n}\eqno(1) $$ where $|\cdot |$ denotes the Lebesgue measure on ${\mathbb{R}}^n$. In 1986 V. Milman {\bf [Mil 1]} discovered that if $B_1$ and $B_2$ are balls there is always a relative position of $B_1$ and $B_2$ for which a perturbed inverse of $(1)$ holds. More precisely:\lq\lq{\sl There exists a constant $C>0$ such that for all $n\in \mathbb{N}$ and any balls $B_1,B_2\subset {\mathbb{R}}^n$ we can find a linear transformation $u:{\mathbb{R}}^n\to {\mathbb{R}}^n$ with $|\det (u)|=1$ and $$|u(B_1)+B_2{|}^{1/n}\le C(|B_1{|}^{1/n}+|B_2{|}^{1/n})"$$} The aim of this paper is to extend this Milman's result to a larger class of sets.

References in corpus (1)

• Factorization theorems for quasi-normed spaces