Abstract
Two functions Δ and Δb, 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, Δb(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. Δb(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 Δb are presented.
Version Information
This revised version was published online in November 2006 with corrections to the Cover Date.
Citation
Victor Klee. David W. Walkup. "The d-step conjecture for polyhedra of dimension d<6." Acta Math. 117 53 - 78, 1967. https://doi.org/10.1007/BF02395040
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