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

Author: Victor Klee; David W Walkup; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB. Ft. Belvoir Defense Technical Information Center DEC 1965. Print book : English WorldCat Two functions A and B, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. A(d, n) is the maximum diameter of convex polyhedra of dimension d with n faces of dimension d-1; similarly, B(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 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. B(d,2d) = d, is true for d

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Document Type: Book Victor Klee; David W Walkup; BOEING SCIENTIFIC RESEARCH LABS SEATTLE WASH MATHEMATICS RESEARCH LAB. Find more information about: Victor Klee David W Walkup 227397643 Also available from the authors. 47 pages

### Abstract:

Two functions A and B, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. A(d, n) is the maximum diameter of convex polyhedra of dimension d with n faces of dimension d-1; similarly, B(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 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. B(d,2d) = d, is true for d <or 5. It is also shown that the general d-step conjecture, i.e. A(d,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.

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