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

A Continuous d-Step Conjecture for Polytopes

“…Following the analogy with the diameter, let Λ(d, n) be the largest total curvature λ(P ) of the central path over all polytopes P defined by n inequalities in dimension d. Klee and Walkup [9] showed that for the Hirsch conjecture the special case where the number of inequalities is twice the dimension is equivalent to the general case. We prove that the same holds for the continuous Hirsch conjecture.…”

“…In [3] the authors showed that a redundant Klee-Minty d-cube C satisfies λ(C) ≥ ( d , and in [4] provided a family of d-dimensional polytopes P defined by n > 2d non-redundant inequalities satisfying lim inf n→∞ λ(P)/n ≥ π for a fixed d. In other words, the continuous analogue of the result of Holt and Klee holds. It has been shown by Klee and Walkup [9] that the special case n = 2d for all d is equivalent to the conjecture of Hirsch. We prove a continuous analogue of the result of Klee and Walkup, namely, we show in Section 2 that Conjecture 1.2 and Conjecture 1.3 are equivalent.…”

Diameter and Curvature: Intriguing Analogies

“…Klee and Walkup [12] showed that the d-step conjecture is equivalent to the conjecture of Hirsch. The continuous analogue of the result of Klee and Walkup.…”

A counterexample to the Hirsch Conjecture

“…In modern terminology, a bounded one is a (convex ) polytope while a perhaps-unbounded one is a polyhedron. But the unbounded case was disproved by Klee and Walkup [34] in 1967 with the construction of a polyhedron of dimension four with eight facets and diameter 5. Since then the expression "Hirsch Conjecture" has been used referring to the bounded case.…”