中文摘要：

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Rn上任意凸体K与L且i=0,…,n-1,是否成立（Wi（K＋L））/（Wi＋1（K＋L））≥（Wi（K））/（Wi＋1（K））＋（Wi（L））/（Wi＋1（L））？这里Wi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Rn上的星体,n-2≤i≤n-1且i∈R,则（Wi（K＋L））/（Wi＋1（K＋L））≤（Wi（K））/（Wi＋1（K））＋（Wi（L））/（Wi＋1（L））/（Wi＋1（L））其中Wi表示星体的i次对偶均值积分.

英文摘要：

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 ≤ i≤ n is it true that, for every pair of convex bodies K and L in/Rn one has （Wi（K＋L））/（Wi＋1（K＋L））≥（Wi（K））/（Wi＋1（K））＋（Wi（L））/（Wi＋1（L））？Here, Wi 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 Rn and if n-2≤i≤n-1, then （Wi（K＋L））/（Wi＋1（K＋L））≤（Wi（K））/（Wi＋1（K））＋（Wi（L））/（Wi＋1（L））/（Wi＋1（L）） where Wi is the i-th dual quermassintegral of a star body.