Log-optimal (d+2)-configurations in d-dimensions

TL;DR

This paper classifies all stationary logarithmic configurations of d+2 points on a sphere in d-dimensions, identifying new minimal energy configurations involving orthogonal simplexes, expanding understanding of optimal point arrangements.

Contribution

It provides a complete classification of stationary logarithmic configurations of d+2 points on the (d-1)-sphere, revealing new minimal energy configurations involving orthogonal simplexes.

Findings

Logarithmic energy minima occur at configurations with two orthogonal regular simplexes.

Global minima are achieved when the simplexes are balanced in size, depending on the parity of d.

The study introduces a new class of energy-minimizing configurations beyond known regular simplex and cross polytope.

Abstract

We enumerate and classify all stationary logarithmic configurations of d+2 points on the unit (d-1)-sphere in d-dimensions. In particular, we show that the logarithmic energy attains its relative minima at configurations that consist of two orthogonal to each other regular simplexes of cardinality m and n. The global minimum occurs when m=n if d is even and m=n+1 otherwise. This characterizes a new class of configurations that minimize the logarithmic energy on the (d-1)-sphere for all d. The other two classes known in the literature, the regular simplex and the cross polytope, are both universally optimal configurations.