Polytopes and arrangements: Diameter and curvature

By analogy with the conjecture of Hirsch, we conjecture that the order of the largest total curvature of the central path associated to a polytope is the number of inequalities deflning the polytope. By analogy with a result of Dedieu, Malajovich and Shub, we conjecture that the average diameter of a bounded cell of an arrangement is less than the dimension. We prove continuous analogues of two results of Holt-Klee and Klee-Walkup: we construct a family of polytopes which attain the conjectured order of the largest total curvature, and we prove that the special case where the number of inequalities is twice the dimension is equivalent to the general case. We substantiate these conjectures in low dimensions and highlight additional links. 1 Continuous Analogue of the Conjecture of Hirsch Let P be a full dimensional convex polyhedron deflned by m inequalities in dimension n. The diameter –(P ) is the smallest number such that any two vertices of the polyhedron P can be connected by a path with at most –(P ) edges. The conjecture of Hirsch, formulated in 1957 and reported in [2], states that the diameter of a polyhedron deflned by m inequalities in dimension n is not greater than m¡n. The conjecture does not hold for unbounded polyhedra. A polytope is a bounded polyhedron. No polynomial bound is known for the diameter of a polytope. Conjecture 1.1. (Conjecture of Hirsch for polytopes) The diameter of a polytope deflned by m inequalities in dimension n is not greater than m ¡ n. Intuitively, the total curvature [15] is a measure of how far ofi a certain curve is from being a straight line. Let ˆ : [fi; fl] ! R be a C((fi ¡ "; fl + ")) map for some " > 0 with a non-zero derivative in [fi; fl]. Denote its arc length by l(t) = R t fi k _̂ (¿)kd¿ , its parametrization by the arc length by ˆarc = ˆ – l¡1 : [0; l(fl)] ! R, and its curvature at the point t by •(t) = ̃̂arc(t). The total curvature is deflned as R l(fl) 0 k•(t)kdt. The requirement _̂ 6= 0 insures that any given segment of the curve is traversed only once and allows to deflne a curvature at any point on the curve. We present one useful proposition. Roughly speaking, it states that two similar curves might not difier greatly in their total curvatures either. This fact is used in Section 3 in proving the analogue of the d-step conjecture for the total curvature of the central path. Proposition 1.1. Let ˆ be as above and f`gj=0;1;::: be a sequence of C((fi ¡ "; fl + ")) functions with non-zero derivatives in [fi; fl] that converge to ˆ point-wise as j ! 1, i.e., `(t) ! ˆ(t) for all t 2 [fi; fl]. Then the total curvature of ˆ is bounded from above by the inflmum limit of the total curvature of ` over all j.