- Acta Math.
- Volume 117 (1967), 53-78.
The d-step conjecture for polyhedra of dimension d<6
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.
This revised version was published online in November 2006 with corrections to the Cover Date.
Acta Math., Volume 117 (1967), 53-78.
Received: 5 April 1966
First available in Project Euclid: 31 January 2017
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1967 © Almqvist & Wiksells Boktryckeri AB
Klee, Victor; Walkup, David W. The d -step conjecture for polyhedra of dimension d <6. Acta Math. 117 (1967), 53--78. doi:10.1007/BF02395040. https://projecteuclid.org/euclid.acta/1485889495