Résumé
The Borsuk conjecture and the V & aacute;zsonyi problem are two attractive and famous questions in discrete and combinatorial geometry, both based on the notion of diameter of bounded sets. In this paper, we present an equivalence between the critical sets with Borsuk number 4 in R3 and the minimal structures for the V & aacute;zsonyi problem by using the well-known Reuleaux polyhedra. The latter leads to a full characterization of all finite sets in R3 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 & aacute;zsonyi problem and showed that the diameter graph arising from involutive polyhedra is vertex (and edge) 4-critical. (c) 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).