Polytope pairs and their relationship to linear programming

Abstract : An important development in the theory of (convex) polytopes was the determination by Barnette and McMullen of the minimum and maximum of v(P) (number of vertices of P) as P ranges over all simple d-polytopes with n facets. Their results are here extended to certain pairs consisting of a polytope and one of its facets. Corollaries of our main results are the determination of the minimum and maximum of v(P) as P ranges over all simple d-polyhedra with u unbounded and n - u bounded facets, and of the minimum and maximum of v(P not F)/v(F) as (P,F) ranges over all pairs consisting of a simple d-polytope P with n facets and a facet F intersecting all other facets of P. Such pairs, called Kirkman pairs of class (d,n), are related to several aspects of linear programming, including a recent algorithm of Mattheiss for finding all vertices of a polytope defined by a system of linear inequalities. (Author)

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