对偶均值积分的Marcus-Lopes不等式

Milman曾提出过一个问题;在混合体积理论,是否存在Marcus-Lopes型和Bergstrom型不等式?即对R~n上任意凸体K与L且i=0,…,n-1,是否成立(W_i(K+L))/(W_i+1(K+L))≥(W_i(K))/(W_i+1(K))+(W_i(L))/(W_i+1(L))?这里W_i表示凸体的i次均值积分.当且仅当i=n-1或i=n-2时,这个问题是正确的,已被证明.作者考虑了一个对偶问题,证明了:若K与L是R~n上的星体,n-2≤i≤n-1且i∈R,则(W_i(K+L))/(W_i+1(K+L))≤(W_i(K))/(W_i+1(K))+(W_i(L))/(W_i+1(L))/(W_i+1(L))其中W_i表示星体的i次对偶均值积分.

On the Marcus-Lopes Inequalities for Dual Quermassintegrals

The main aim of this paper is a question of Milman about a possible analogue of the Marcus-Lopes inequality and Bergstrom's inequality in the theory of mixed volumes: for which values of $0\leq i\leq n$ is it true that, for every pair of convex bodies $K$ and $L$ in ${\Bbb R^{n}}$ one has $$\frac{W_{i}(K+L)}{W_{i+1}(K+L)}\geq \frac{W_{i}(K)}{W_{i+1}(K)}+\frac{W_{i}(L)}{W_{i+1}(L)}? $$ Here, $W_{i}$ is the $i$-th quermassintegral of a convex body. The answer to this question was proved to be positive if and only if $i=n-1$ or $i=n-2$. In this paper, the author proves an analogous statement for the dual quermassintegrals. If $K$ and $L$ are star bodies in ${\Bbb R^{n}}$ and if $n-2\leq i \leq n-1$, then $$\frac{\wt{W}_{i}(K\wt{+}L)}{\wt{W}_{i+1}(K\wt{+}L)}\leq \frac{\wt{W}_{i}(K)}{\wt{W}_{i+1}(K)}+\frac{\wt{W}_{i}(L)}{\wt{W}_{i+1}(L)}, $$ where $\wt{W}_{i}$ is the $i$-th dual quermassintegral of a star body.