球凸体的西尔维斯特四点问题

中文摘要 AI 辅助

我们研究球凸体的经典西尔维斯特四点问题，其中凸包被固定半径R的球的交集所替代。我们证明Efron-Buchta恒等式在该框架下依然成立，因此n个均匀随机点的R-球凸包的顶点数分布由其子样本凸包体积的矩决定，反之亦然。对于任意平面球凸体中的三个和四个均匀随机点，我们确定了R-球凸包的西尔维斯特概率。我们还给出了Groemer不等式的推广，用于计算n个随机点的球凸包的期望面积，并证明该面积由相同面积的圆盘最小化。特别地，这给出了边界曲线仿射长度的球凸类似物的等周不等式。对于单位圆盘，我们可以用半径R的双对数函数表示西尔维斯特概率，并给出R=1时的显式值。当R→∞时，这些概率恢复经典西尔维斯特概率。作为另一个结果，我们确定了圆盘上三个均匀随机点的外接圆半径在[1,∞)上的精确分布函数。

英文摘要

We investigate Sylvester's classical four-point problem for ball-convex bodies, where convex hulls are replaced by intersections of balls of a fixed radius $R$. We show that the Efron--Buchta identities persist in this framework, so that the distribution of the number of vertices of the $R$-ball-convex hull of $n$ uniform random points is determined by the moments of the volumes of the hulls of its subsamples, and vice versa. For three and four uniform random points in an arbitrary planar ball-convex body, we determine the Sylvester probabilities of the $R$-ball-convex hull. We also give an extension of Groemer's inequality for the expected area of the ball-convex hull of $n$ random points and show that it is minimised by the disc of the same area. In particular, this yields an isoperimetric inequality for the ball-convex analogue of the affine length of the boundary curve. For the unit disc, we can express the Sylvester probabilities using dilogarithmic functions of the radius $R$, and give explicit values at $R=1$. As $R\to\infty$, these recover the classical Sylvester probabilities. As another consequence, we determine the exact distribution function of the circumradius of three uniform random points in he disc on $[1,\infty)$.

发表机构

• University of Szeged（塞格德大学）
• Technische Universität Wien（维也纳工业大学）

机构由 AI 辅助整理，请以论文原文为准。