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.
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References
Balinski, M., On the graph structure of convex polyhedra inn-space.Pacific J. Math.,11 (1961), 431–434.
Dantzig, G. B.,Linear programming and extensions. Princeton Univ. Press, Princeton, N. J., 1963.
— Eight unsolved problems from mathematical programming.Bull. Amer. Math. Soc., 70 (1964), 499–500.
Eggleston, H. G., Grünbaum, B. &Klee, V., Some semicontinuity theorems for convex polytopes and cell-complexes.Comment. Math. Helv., 39 (1964), 165–188.
Gale, D., Neighborly and cyclic polytopes.Proceedings of Symposia in Pure Math., vol. 7, Convexity, Amer. Math. Soc., Providence, R.I., 1963, pp. 225–232.
— On the number of faces of a convex polytope.Canad. J. Math., 16 (1964), 12–17.
Grünbaum, B. &Motzkin, T. S., Longest simple paths in polyhedral graphs.J. London Math. Soc., 37, (1962), 152–160.
— On polyhedral graphs.Proceedings of Symposia in Pure Math., vol. 7, Convexity, Amer. Math. Soc., Providence, R.I., 1963, pp. 285–290.
Klee, V., Diameters of polyhedral graphs.Canad. J. Math., 16 (1964), 602–614.
—, On the number of vertices of a convex polytope.Canad. J. Math., 16 (1964), 701–720.
— Convex polytopes and linear programming.Proceedings of the IBM Scientific Computing Symposium on Combinatorial Problems, March 1964. International Business Machines Corporation, Data Processing Division, White Plains, New York, 1966, pp. 123–158.
— Paths on polyhedra. I.J. Soc. Indust. Appl. Math., 13 (1965), 946–956.
— Paths on polyhedra. II.Pacific J. Math., 17 (1966), 249–262.
Steinitz, E. &Rademacher, H.,Vorlesungen über die Theorie der Polyeder. Springer, Berlin, 1934.
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Klee, V., Walkup, D.W. Thed-step conjecture for polyhedra of dimensiond<6. Acta Math. 117, 53–78 (1967). https://doi.org/10.1007/BF02395040
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DOI: https://doi.org/10.1007/BF02395040