Diameter and Curvature: Intriguing Analogies

We highlight intriguing analogies between the diameter of a polytope and the largest possible total curvature of the associated central path. We prove continuous analogues of the results of Holt and Klee, and Klee and Walkup: We construct a family of polytopes which attain the conjectured order of the largest curvature, and prove that the special case where the number of inequalities is twice the dimension is equivalent to the general case. We show that the conjectured bound for the average diameter of a bounded cell of a simple hyperplane arrangement is asymptotically tight for fixed dimension. Links with the conjecture of Hirsch, Haimovichʹs probabilistic analysis of the shadow-vertex simplex algorithm, and the result of Dedieu, Malajovich and Shub on the average total curvature of a bounded cell are presented.