论文标题
最大表面积多面体，带有五个顶点，$ \ mathbb {s}^2 $
The Maximum Surface Area Polyhedron with Five Vertices Inscribed in the Sphere $\mathbb{S}^2$
论文作者
论文摘要
本文重点介绍了分析确定单位球上五个点的最佳位置的问题，$ \ mathbb {s}^2 $，以便最大化点的凸面船体的表面积。结果表明，最佳多面体具有三角双锥体结构，其中两个顶点位于北极和南极，其他三个顶点构成了赤道中刻有的等边三角形。该结果证实了Akkiraju的猜想，他对最大化器进行了数值搜索。作为晶体学的应用，表面积差异被认为是观察到的配位多面体与理想的量度之间的失真度量。主要结果产生了一个具有五个顶点的配位多面体表面积差异的公式。
This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere $\mathbb{S}^2$ so that the surface area of the convex hull of the points is maximized. It is shown that the optimal polyhedron has a trigonal bipyramidal structure with two vertices placed at the north and south poles and the other three vertices forming an equilateral triangle inscribed in the equator. This result confirms a conjecture of Akkiraju, who conducted a numerical search for the maximizer. As an application to crystallography, the surface area discrepancy is considered as a measure of distortion between an observed coordination polyhedron and an ideal one. The main result yields a formula for the surface area discrepancy of any coordination polyhedron with five vertices.
