Sections and projections of convex polytopes

Abstract

Introduction . Gale diagrams, and similar techniques, have been used in several recent investigations into the combinatorial properties of convex polytopes. Here we describe other applications, namely to problems concerning sections and projections of polytopes of a given combinatorial type. For example, in §4 we shall show that every n –dimensional convex polytope with at most n + 2 vertices possesses the universal shadow boundary (usb) property : If is a subcomplex of the boundary complex ℬ ( P ) of P , and set is homeomorphic to an ( n – 2)-sphere, then there exists a polytope P', combinatorially isomorphic to P , such that the complex corresponding to is a shadow boundary of P' . We shall also show that this is the best possible result in that polytopes with n + 3 or more vertices do not, in general, have the usb property. A second application of the method, to be described in §5, is to the construction of non-realisable complexes.