Borsuk and V\'azsonyi problems through Reuleaux polyhedra

TL;DR

This paper establishes a connection between Borsuk and Vázquez problems in 3D geometry using Reuleaux polyhedra, leading to a complete characterization of finite sets with Borsuk number 4.

Contribution

It introduces an equivalence between critical sets with Borsuk number 4 and minimal structures for the Vázquez problem, proving a related conjecture and characterizing diameter graphs.

Findings

Proved a conjecture on strongly critical configurations for Vázquez problem.

Established that diameter graphs from involutive polyhedra are vertex and edge 4-critical.

Provided a full characterization of finite sets in R^3 with Borsuk number 4.

Abstract

The Borsuk conjecture and the V\'azsonyi problem are two attractive and famous questions in discrete and combinatorial geometry, both based on the notion of diameter of a bounded sets. In this paper, we present an equivalence between the critical sets with Borsuk number 4 in $\mathbb{R}^3$ and the minimal structures for the V\'azsonyi problem by using the well-known Reuleaux polyhedra. The latter lead to a full characterization of all finite sets in $\mathbb{R}^3$ with Borsuk number 4. The proof of such equivalence needs various ingredients, in particular, we proved a conjecture dealing with strongly critical configuration for the V\'azsonyi problem and showed that the diameter graph arising from involutive polyhedra is vertex (and edge) 4-critical.