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References
Barnette, D. W., The minimum number of vertices of a simple polytope.Israel J. Math., 10 (1971), 121–125.
— Graph theorems for manifolds.Israel J. Math., 16 (1973), 62–72.
Brown, W. G., Historical note on a recurrent combinatorial problem.Amer. Math. Monthly, 72 (1965), 973–977.
Bruggesser, H. &Mani, P., Shellable decompositions of cells and spheres.Math. Scand., 29 (1972), 197–205.
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.
Gale, D., Neighboring vertices on a convex polyhedron.Linear Inequalities and Related Systems (H. W. Kuhn and A. W. Tucker, eds.), Ann. of Math. Studies 38 (1956), 255–263, Princeton Univ. Press, Princeton, N.J.
— Neighborly and cyclic polytopes.Convexity (V. Klee, ed.), Proc. Symp. Pure Math., 7 (1963), 255–232, Amer. Math. Soc., Providence, T.I.
— On the number of faces of a convex polytope.Canadian J. Math., 16 (1964), 12–17.
Grünbaum, B.,Convex Polytopes. Interscience-Wiley, London, 1967.
Grünbaum, B. &Sreedharan, V., An enumeration of simplicial 4-polytopes with 8 vertices.J. Combinatorial Theory, 2 (1967), 437–465.
Kirkman, T. P., On the enumeration ofX-edra having triedal summits and an (x−1)-gonal base.Philos. Trans. Royal Soc. London Ser A, 146 (1856), 399–411.
— On the partitions of theR-pyramid, being the first class ofR-gonousX-edra.Philos. Trans. Royal. Soc. London Ser. A, 148 (1858), 145–161.
Klee, V., Diameters of polyhedral graphs.Canadian J. Math., 16 (1964), 602–614.
— On the number of vertices of a convex polytope.Canadian J. Math., 16 (1964), 701–720.
— Convex polytopes and linear programming.Procedings of the IBM Scientific Computing Symposium on Combinatorial Problems March 16–18, 1964, 123–158, IBM Data Processing Division, White Plains, N.Y., 1966.
— A comparison of primal and dual methods for linear programming.Numer. Math., 9 (1966), 227–235.
Klee, V. &Minty, G. J., How good is the simplex algorithm?Inequalities III (O. Shisha, ed.), 159–175. Academic Press, N. Y., 1972.
Klee, V. &Walkup, D. W., Thed-step conjecture for polyhedra of dimensiond<6.Acta Math., 117 (1967), 53–78.
Maňas, M. &Nedoma, J., Finding all vertices of a convex polyhedron.Numer. Math., 12 (1968), 226–229.
Mattheiss, T. H., An algorithm for determining irrelevant constraints and all vertices in systems of linear inequalities.Operations Res., 21 (1973), 247–260.
McMullen, P., The maximum number of faces of a convex polytope.Mathematika, 17 (1960), 179–184.
Motzkin, T. S., Cooperative classes of finite sets in one and more dimensions.J. Combinatorial Theory, 3 (1967), 244–251.
Motzkin, T. S. &O'Neil, P. E., Bounds assuring subsets in convex position.J. Combinatorial Theory, 3 (1967), 252–255.
Ordman, E., Algebraic characterization of some classical combinatorial problems.Amer. Math. Monthly, 78 (1971), 961–970.
Rademacher, H., On the number of certain types of polyhedra.Illinois J. Math., 9 (1965), 361–380.
Sallee, G. T., Incidence graphs of convex polytopes.J. Combinatorial Theory, 2 (1967), 466–506.
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Klee, V. Polytope pairs and their relationship to linear programming. Acta Math. 133, 1–25 (1974). https://doi.org/10.1007/BF02392139
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DOI: https://doi.org/10.1007/BF02392139