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Borsuk and V\'azsonyi problems through Reuleaux polyhedra
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 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
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