Borsuk and Vázsonyi problems through Reuleaux polyhedra

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 $\mathbb{R}<sup>3$ 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 $\mathbb{R}<sup>3$ 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.