Borsuk and Vázsonyi problems through Reuleaux polyhedra

Abstract

The Borsuk conjecture and the Vázsonyi 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 Vázsonyi 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 Vázsonyi problem and showed that the diameter graph arising from involutive polyhedra is vertex (and edge) 4-critical.