Twenty Problems on Convex Polyhedra Part II

Abstract

Let P be a given convex polyhedron and L any plane. Then if L ∩ P is not empty it is called a plane section of P . Every plane section of P is either a convex polygon, an edge or a vertex of P . VI. If a convex polyhedron P has v vertices (v ≥4), does there always exist a plane section of P which is a convex n-gon with n ≥ √ v ? A more general statement of this problem is to ask whether there exists a constant d (0 < d < 1) such that every convex polyhedron P with v vertices possesses a plane section which is an n -gon with n ≥ v d . Problem VI is then equivalent to asking whether d ≤ ½