The maximum surface area polyhedron with five vertices inscribed in the sphere

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...

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Bibliographic Details
Published in:Acta crystallographica. Section A, Foundations and advances Foundations and advances, 2021-01, Vol.77 (1), p.67-74
Main Authors: Donahue, Jessica, Hoehner, Steven, Li, Ben
Format: Article
Language:English
Subjects:
Apexes
Convexity
Coordination
Crystallography
Equator
inequalities
optimization
Polyhedra
Surface area
Triangles
triangular bipyramids
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Summary: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. It is shown that a polyhedron with the trigonal bipyramidal structure is the unique surface area maximizer among all polyhedra with five vertices inscribed in a sphere.
ISSN:2053-2733
0108-7673
2053-2733
DOI:10.1107/S2053273320015089