A Continuous d-Step Conjecture for Polytopes

Abstract

The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as the continuous analogue of its diameter. We prove the analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytope defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.

An Analogue of the Klee–Walkup Result for Sonnevend’s Curvature of the Central Path

“…The following proposition, first proved in [3], shows that this is indeed the case. Proof Claim 1 is the same as Proposition 2.1 in [3] (see also the remark following it). Statement 2. follows from the first part sinces n+1 (μ)…”

“…An analogous behavior is known for both the diameter of a polytope [4] and the total geometric curvature of the central path [3,9,10]. The lower bound Ω(n) for the Sonnevend curvature is also analogous to the worst-case lower bound known for the geometric curvature [3,9,10].…”

“…In this section, we introduce the construction used in [3]. Using this construction, our main result for the Sonnevend curvature…”

“…In that paper, the authors construct a sequence of polytopes, whose central path approximates that of the previous one. Moreover, the new central path has an additional extra "sharp turn," and it follows that [3] the total geometric curvature of the new central path increases by a constant. In our paper, we also use their construction and we investigate it further aiming to analyze the behavior of the Sonnevend curvature on an interval corresponding to a neighborhood of that sharp turn.…”

“…Our work is motivated by the work of Deza et al [3] about the geometric curvature of the central path. In that paper, the authors construct a sequence of polytopes, whose central path approximates that of the previous one.…”