This article focuses on the problem of analytically determining the optimal placement of five points on the unit sphere 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.

中文翻译：

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球面上刻有五个顶点的最大表面积多面体

本文着重于分析确定单位球面上五个点的最佳位置的问题，以使这些点的凸包的表面积最大化。结果表明，最佳多面体具有三角双锥结构，两个顶点分别位于南极和北极，而其他三个顶点形成赤道内切的等边三角形。这个结果证实了Akkiraju的猜想，他对最大值最大化进行了数值搜索。作为晶体学的一种应用，表面积差异被认为是观察到的配位多面体与理想多面体之间变形的一种量度。主要结果得出了具有五个顶点的任何配位多面体的表面积差异的公式。