On a conjecture of A. Bezdek
In 1985, V. I. Proizvolov posed the question whether any \(n\) points \(A_1\dots,A_n\) inside a convex \(n\)-gon with sides \(s_1,\dots,s_n\) can be renumbered such that the triangles with bases \(s_i\) and a vertex \(A_i\) cover the given \(n\)-gon. A. Bezdek proved a generalization of this, and the author of the present paper verifies a related conjecture of A. Bezdek by showing the following: Let \(k\) points \(a_1,\dots,a_k\in R^n\) and a convex polytope \(P\subset R^n\) be given. Then one can choose \(k\) facets \(F_1,\dots,F_k\) of \(P\) such that the convex hulls \(\text{conv}(\{a_i\}\cup F_i)\) either cover the polytope \(P\) or do not pairwise intersect in interior points. Some further statements (referring to polyhedral sets, i.e., to possibly unbounded intersections of halfspaces) are proved, too. At the end of the article some interesting related conjectures and questions are given.
scientific article; zbMATH DE number 1747951