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 defining 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 defined 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...

• Type: Journal
• Year: 2008
• Where: ORL
• Authors: Antoine Deza, Tamás Terlaky, Yuriy Zinchenko
• Keywords: Conjecture, Largest Total Curvature, ORL 2008, Polytope