Volume product of planar polar convex bodies --- lower estimates with stability
Volume product of planar polar convex bodies --- lower estimates with stability
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平面极凸体的体积乘积——较低的估计值且稳定
DOI：
10.1556/sscmath.50.2013.2.1235
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
S. Reisner
中科院分区：
文献类型：
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作者：
K. Böröczky;E. Makai;M. Meyer;S. Reisner
设$K \subset {\mathbb R}^2$是一个o$-对称凸体，$K^*$是它的极体.我们有$|K|\cdot| K^*|\ge 8$，相等当且仅当$K$是一个整数。($|\c点|$表示体积）。如果$K \subset {\mathbb R}^2$是凸体，其中$o \in {\text{int}}\，K$，则$|K|\c点|K^*|\ge 27/4$，等式当且仅当$K$是三角形且$o$是它的质心。如果$K \subset {\mathbb R}^2$是凸体，则我们有$|K|\cdot| [（K-K）/2）]^*|\ge 6$，等式当且仅当$K$是三角形。这些定理分别是由马勒和赖斯纳、马勒和迈耶以及埃格莱斯顿提出的。我们证明了一个类似的定理：如果$K$关于$o$具有$n$重旋转对称，则$|K|\cdot| K^*|\geN^2\sin ^2（\pi /n）$，相等当且仅当$K$是以$o$为中心的正则$n$-边形。我们还将给出这四个不等式的稳定性变式，包括物体和极性中心。为此，我们使用巴拿赫-马祖尔距离（从平行四边形，或三角形），或其类似的副本，而不是仿射变换（从规则$n$-n），分别。稳定性变量是尖锐的，直到常数因子。我们推广了不等式$|K|\cdot| K^*|\ge n^2 \sin ^2（\pi /n）$到具有$o \in {\text{int}}\，K$的物体，它们包含并且被包含在两个正则$n$-边形中，包含的$n$-边形的顶点与包含的$n$-边形的边相关联。我们的关键引理是一个稳定性估计的面积积的两个部门的凸体极对方。对于我们的几个陈述，我们给出了几个证明;特别是，我们给出了马勒-赖斯纳定理的一个新证明。
Let $K \subset {\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\cdot |K^*| \ge 8$, with equality if and only if $K$ is a parallelogram. ($| \cdot |$ denotes volume). If $K \subset {\mathbb R}^2$ is a convex body, with $o \in {\text{int}}\,K$, then $|K|\cdot |K^*| \ge 27/4$, with equality if and only if $K$ is a triangle and $o$ is its centroid. If $K \subset {\mathbb R}^2$ is a convex body, then we have $|K| \cdot |[(K-K)/2)]^* | \ge 6$, with equality if and only if $K$ is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if $K$ has $n$-fold rotational symmetry about $o$, then $|K|\cdot |K^*| \ge n^2 \sin ^2 ( \pi /n)$, with equality if and only if $K$ is a regular $n$-gon of centre $o$. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular $n$-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality $|K|\cdot |K^*| \ge n^2 \sin ^2 ( \pi /n)$ to bodies with $o \in {\text{int}}\,K$, which contain, and are contained in, two regular $n$-gons, the vertices of the contained $n$-gon being incident to the sides of the containing $n$-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.
关于对称凸体的体积积
DOI：
--
发表时间：
2023
期刊：
影响因子：
--
作者：
Sakata N.;Mishina R.;Ogawa M.;Ishihara K.;Koda Y.;Ozawa M.;Shimokawa K.;Hiroshi Iriyeh
通讯作者：
Hiroshi Iriyeh