The d-step conjecture for polyhedra of dimension d<6

Abstract

Two functions Δ and Δ_b, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. Δ(d, n) is the maximum diameter of convex polyhedra of dimension d with n faces of dimension d−1; similarly, Δ_b(d,n) is the maximum diameter of bounded polyhedra of dimension d with n faces of dimension d−1. The diameter of a polyhedron P is the smallest integer l such that any two vertices of P can be joined by a path of l or fewer edges of P. It is shown that the bounded d-step conjecture, i.e. Δ_b(d,2d)=d, is true for d≤5. It is also shown that the general d-step conjecture, i.e. Δ(d, 2d)≤d, of significance in linear programming, is false for d≥4. A number of other specific values and bounds for Δ and Δ_b are presented.