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

Abstract

We determine the optimal placement of five points on the unit sphere S 2 so that the surface area of the convex hull of the points is maximized. We show that the surface area is maximized by a triangular bipyramid with two points placed at the north and south poles and the other three points forming an equilateral triangle inscribed in the equator.