Five Essays on the Geometry of L\'aszl\'o Fejes T\'oth

TL;DR

This paper explores key geometric problems studied by László Fejes Tóth, including circle packings, sphere point distributions, kissing numbers, and isoperimetric problems for polyhedra, highlighting their mathematical significance.

Contribution

The paper provides a comprehensive analysis of Fejes Tóth's contributions to various fundamental problems in discrete and computational geometry.

Findings

Bounds on circle packings and sphere packings

Maximization of minimum distances on spheres

Results on maximum kissing numbers and isoperimetric problems

Abstract

In this paper we consider the following topics related to results of L\'aszl\'o Fejes T\'oth: (1) The Tammes problem and Fejes T\'oth's bound on circle packings; (2) Fejes T\'oth's problem on maximizing the minimum distance between antipodal pairs of points on the sphere; (3) Fejes T\'oth's problem on the maximum kissing number of packings on the sphere; (4) The Fejes T\'oth -- Sachs problem on the one--sided kissing numbers; (5) Fejes T\'oth's papers on the isoperimetric problem for polyhedra.