| 马统一.再论仿射不变量$W_{i}(K)W_{i}(K^{})$的下界估计[J].数学年刊A辑,2013,34(6):747~760 |
| 再论仿射不变量$W_{i}(K)W_{i}(K^{})$的下界估计 |
| On Lower Bound for Affine Invariant$W_{i}(K)W_{i}(K^{})$ |
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| DOI： |
| 中文关键词: 凸体, 混合体积, $L_{p}$-\!\!曲率映象, $L_{p}$-John椭球, Mahler猜想 |
| 英文关键词:Convex body, Mixed volumes, $L_{p}$-Curvature image, $L_{p}$-John ellipsoid, Mahler conjecture |
| 基金项目:国家自然科学基金 (No.11161019) |
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| 中文摘要: |
| 对于所有凸体与每一个$i$, 寻找仿射不变量$W_{i}(K)W_{i}(K^{*})$下界的问题是一个至今未能完全解决的公开问题.
最近,赵长健考虑了仿射不变量$W_{i}(K)W_{i}(K^{*})$的下界是与凸体$K$本身有关的常数的情形, 并利用混合体积与对偶混合体积的关系理论,
对仿射不变量$W_{i}(K)W_{i}(K^{*})$的下界进行了讨论. 本文进一步讨论仿射不变量$W_{i}(K)W_{i}(K^{*})$的下界估计,
并对具有正的连续曲率且包含原点为其内点的凸体\!$K$, 获得了仿射不变量$W_{i}(K)W_{i}(K^{*})$的几个不同精度的下界, 同时给出了著名的Bourgain-Milman
不等式中通用常数$c$的具体表示值.最后提出了两个公开问题. |
| 英文摘要: |
| The problem of finding the lower bound of the product $W_{i}(K)W_{i}(K^{*})$ for all convex bodies for
each $i$, has not been solved completely yet. If the lower bound is related to $K$ itself,
Zhao gave a lower bound of the product $W_{i}(K)W_{i}(K^{*})$ by relation theory of the mixed volume and dual mixed volume.
In this paper,
the author further studies the lower bound of affine invariant $W_{i}(K)W_{i}(K^{*})$,
and gives some lower bounds with different accuracy of the product $W_{i}(K)W_{i}(K^{*})$ for all convex bodies containing the origin
in their interior with a positive continuous curvature function.
Moreover, the specific value of universal constants $c$ for famous Bourgain-Milman inequality is obtained.
Finally, two open problems are presented. |
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