| 赵长健.对偶均值积分的Marcus-Lopes不等式[J].数学年刊A辑,2014,35(4):501~510 |
| 对偶均值积分的Marcus-Lopes不等式 |
| On the Marcus-Lopes Inequalities for Dual Quermassintegrals |
| |
| DOI： |
| 中文关键词: 凸体, 星体,
均值积分, 对偶均值积分, Brunn-Minkowski理论,
对偶Brunn-Minkowski理论 |
| 英文关键词:Convex body, Star body,
Quermassintegral, Dual quermassintegral, Brunn-Minkowski theory,
Dual Brunn-Minkowski theory |
| 基金项目:国家自然科学基金 (No.11371334) |
|
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| 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${\Bbb R^{n}}$上任意凸体$K$与$L$且$i=0,\cdots,n-1$, 是否成立
$$
\frac{W_{i}(K+L)}{W_{i+1}(K+L)}\geq
\frac{W_{i}(K)}{W_{i+1}(K)}+\frac{W_{i}(L)}{W_{i+1}(L)}?
$$
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${\Bbb
R^{n}}$上的星体, $n-2\leq i\leq n-1$且$i\in {\Bbb R}$,则
$$
\frac{\wt{W}_{i}(K\wt{+}L)}{\wt{W}_{i+1}(K\wt{+}L)}\leq
\frac{\wt{W}_{i}(K)}{\wt{W}_{i+1}(K)}+\frac{\wt{W}_{i}(L)}{\wt{W}_{i+1}(L)},
$$
其中
$\wt{W}_{i}$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 $0\leq i\leq n$ 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. |
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