Spherical Distribution of 5 Points with Maximal Distance Sum

Abstract

In this paper, we consider the problem of spherical distribution of 5 points, that is, how to configure 5 points on the unit sphere such that the mutual distance sum is maximal. It is conjectured that the sum of distances is maximal if the 5 points form a bipyramid distribution with two points positioned at opposite poles of the sphere and the other three positioned uniformly on the equator. We study this problem using interval methods and related techniques, and give a computer-assisted proof.

Hilbert’s ‘Monkey Saddle’ and Other Curiosities in the Equilibrium Problem of Three Point Particles on a Circle for Repulsive Power Law Forces

“…So far, using elementary arguments, the regular triangular bi-pyramidal configuration was rigorously shown in [DLT02] to be the unique (up to rotation) optimizer on S 2 of the logarithmic energy (s = 0). A rigorous, computer-assisted proof of the optimality on S 2 of the regular triangular bi-pyramidal configuration if s ∈ (−2, 0) or s ∈ (0, ‫)ש‬ was achieved only recently [Schw16], where it is also proved that the regular triangular bi-pyramidal configuration is not an optimizer on S 2 when s > ‫.ש‬ That the same configuration maximizes the sum of distances on S 2 and therefore minimizes the Riesz −1-energy on S 2 was established earlier with a different computer-aided proof in [HoSh11]. Reference [Schw16] also contains a computer-assisted proof that a square pyramid with s-dependent height is the unique (up to rotation) optimizer on S 2 when ‫ש‬ < s < 15 + 25/512.…”

Distributing many points on spheres: Minimal energy and designs

“…In general, the five point problem is a difficult problem to analyse rigorously. Recently, the papers [179] (for the Coulomb case s = 1 and for s = 2) and [131] (for sum of distances, s = −1) provided computer-assisted proofs that the triangular bi-pyramid is optimal, whereas in the logarithmic case a conventional proof was given in [86]. In [194] a bi-quadratic energy functional is considered.…”

Minimal biquadratic energy of 5 particles on 2-sphere

“…Dragnev, Legg, and Townsend [2] give a solution of the problem for f (r) = − log r known as Whyte's problem. Hou and Shao [3] give a rigorous computeraided solution for f (r) = −r, for which the problem is well-known in discrete geometry. Schwartz [4] gives a rigorous computer-aided solution of Thomson's problem.…”

“…Schwartz [4] gives a rigorous computer-aided solution of Thomson's problem. The results of [3] and [4] involve massive calculations that require a computer. In all these three results, a unique minimizer is the so-called triangular bipyramid (TBP) that consists of two antipodal points, say the North and South poles, and three points on the equator forming an equilateral triangle.…”