The Tammes Problem in $\mathbb{R}^{n}$ and Linear Programming Method

TL;DR

This paper investigates the Tammes problem of optimally arranging points on high-dimensional spheres to maximize minimum distances, using linear programming techniques to identify conditions for optimal configurations and illustrating these with examples.

Contribution

It establishes sufficient conditions for optimal point arrangements on spheres in any dimension using linear programming, expanding understanding of sphere packing configurations.

Findings

Derived conditions for optimal arrangements in various dimensions

Provided examples of optimal configurations

Enhanced understanding of sphere packing structures

Abstract

The Tammes problem delves into the optimal arrangement of $N$ points on the surface of the $n$-dimensional unit sphere (denoted as $\mathbb{S}^{n-1}$), aiming to maximize the minimum distance between any two points. In this paper, we articulate the sufficient conditions requisite for attaining the optimal value of the Tammes problem for arbitrary $n, N \in \mathbb{N}^{+}$, employing the linear programming framework pioneered by Delsarte et al. Furthermore, we showcase several illustrative examples across various dimensions $n$ and select values of $N$ that yield optimal configurations. The findings illuminate the intricate structure of optimal point distributions on spheres, thereby enriching the existing body of research in this domain.