THE D-STEP CONJECTURE FOR POLYHEDRA OF DIMENSION D<6,
BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB
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Two functions A and B, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. Ad,n is the maximum diameter of convex polyhedra of dimension d with n faces of dimension d-1 similarly, Bd,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 k such that any two vertices of P can be joined by a path of k or fewer edges of P. It is shown that the bounded d-step conjecture, i.e. Bd,2d d, is true for d or 5. It is also shown that the general d-step conjecture, i.e. Ad,2d or d, of significance in linear programming, is false for d equal to or greater than 4. A number of other specific values and bounds for A and B are presented.
- Operations Research