Abstract
Adler and Dantzig [2] show that abstract polytopes include simple polytopes as a special case. Further they extend the results of Klee and Walkup [3] to show that the Hirsch conjecture holds for the larger class of abstract polytopes that have dimension less than or equal to five.
It is the purpose of this paper to further extend to abstract polytopes another result from Klee and Walkup [3] which states that the Hirsch conjecture is mathematically equivalent to three other statements. This result makes it possible to look at the Hirsch conjecture by applying the well-defined structure and theorems of abstract polytopes to any of its four equivalent statements.
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References
I. Adler, “Abstract polytopes”, Ph.D. Thesis, Department of Operations Research, Stanford University (1971).
I. Adler and G.B. Dantzig, “Maximum diameter of abstract polytopes”,Mathematical Programming Study 1 (1974) 20–40.
V. Klee and D.W. Walkup, “Thed-step conjecture for polyhedra of dimensiond < 6”,Acta Mathematica 117 (1967) 53–78.
J.A. Lawrence, Jr., “Equivalent formulations of the Hirsch conjecture for abstract polytopes”, ORC Rept. 72-28, Operations Research Center, University of California, Berkeley (1972).
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Lawrence, J.A. Abstract polytopes and the hirsch conjecture. Mathematical Programming 15, 100–104 (1978). https://doi.org/10.1007/BF01609004
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DOI: https://doi.org/10.1007/BF01609004