Acta Mathematica

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

Victor Klee and David W. Walkup

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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.

Note

This revised version was published online in November 2006 with corrections to the Cover Date.

Article information

Source
Acta Math. Volume 117 (1967), 53-78.

Dates
Received: 5 April 1966
First available in Project Euclid: 31 January 2017

Permanent link to this document
http://projecteuclid.org/euclid.acta/1485889495

Digital Object Identifier
doi:10.1007/BF02395040

Zentralblatt MATH identifier
0163.16801

Rights
1967 © Almqvist & Wiksells Boktryckeri AB

Citation

Klee, Victor; Walkup, David W. The d -step conjecture for polyhedra of dimension d &lt;6. Acta Math. 117 (1967), 53--78. doi:10.1007/BF02395040. http://projecteuclid.org/euclid.acta/1485889495.


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