, Volume 117, Issue 1, pp 53-78

Thed-step conjecture for polyhedra of dimensiond<6

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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 dimensiond withn faces of dimensiond−1; similarly, Δ b (d,n) is the maximum diameter of bounded polyhedra of dimensiond withn faces of dimensiond−1. The diameter of a polyhedronP is the smallest integerl such that any two vertices ofP can be joined by a path ofl or fewer edges ofP. It is shown that the boundedd-step conjecture, i.e. Δ b (d,2d)=d, is true ford≤5. It is also shown that the generald-step conjecture, i.e. Δ(d, 2d)≤d, of significance in linear programming, is false ford≥4. A number of other specific values and bounds for Δ and Δ b are presented.

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