标题： Milman逆Brunn-Minkowski不等式的扩展 显示英文标题

标题： An extension of Milman's reverse Brunn-Minkowski inequality

摘要： 经典的Brunn-Minkowski不等式指出，对于$A_1,A_2\subset\R^n$紧集，$$ |A_1+A_2|^{1/n}\ge |A_1|^{1/n}+|A_2|^{1/n}\eqno(1) $$其中$|\cdot|$表示$\R^n$上的勒贝格测度。 1986年，V. Milman{\bf [米勒 1]}发现如果$B_1$和$B_2$是球体，则总存在$B_1$和$B_2$的相对位置，使得$(1)$的扰动逆存在。 更准确地说：\lq \lq{\sl There exists a constant $C>0$ such that for all $n\in\N$ and any balls $B_1,B_2\subset\R^n$ we can find a linear transformation $u\colon\R^n\to\R^n$ with $|{\rm det}(u)|=1$ and $$|u(B_1)+B_2|^{1/n}\le C(|B_1|^{1/n}+|B_2|^{1/n})"$$} 本文的目的是将这一Milman的结果扩展到更大的集合类中。 显示英文摘要

摘要： The classical Brunn-Minkowski inequality states that for $A_1,A_2\subset\R^n$ 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 $\R^n$. 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 $n\in\N$ and any balls $B_1,B_2\subset\R^n$ we can find a linear transformation $u\colon\R^n\to\R^n$ with $|{\rm det}(u)|=1$ 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.