Equivalent Formulations of the Hirsch Conjecture for Abstract Polytopes
CALIFORNIA UNIV BERKELEY OPERATIONS RESEARCH CENTER
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Abstract polytopes are mathematical creations which are defined by three axioms. It has been shown that simple polytopes are a proper subclass of abstract polytopes. Hence theorems proving facts about abstract polytopes in general, prove facts about simple polytopes in particular. Klee and Walkup showed the following four statements were mathematically equivalent for simple polytopes Any two vertices of a simple polytope can be joined by a W sub v nonreturning path. Deltan,d or n - d Hirsch conjecture. Delta2d,d or d . For a Dantzig figure, P,x,y , delta sub p x,y d . The purpose of the paper is to show that the four statements above are equivalent for the larger class of abstract polytopes as well.
- Operations Research