The Maximum Surface Area Polyhedron with Five Vertices Inscribed in the Sphere $\mathbb{S}^2$

TL;DR

This paper analytically determines the optimal arrangement of five points on a sphere to maximize the convex hull's surface area, confirming a conjecture and providing formulas relevant to crystallography.

Contribution

It proves the optimal configuration of five points on a sphere for maximum surface area and confirms a previous numerical conjecture.

Findings

Optimal polyhedron is a trigonal bipyramid with vertices at poles and an equilateral triangle on the equator.

Derived a formula for surface area discrepancy in five-vertex coordination polyhedra.

Confirmed the conjecture of Akkiraju regarding the maximizer configuration.

Abstract

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.