Diameter and Curvature: Intriguing Analogies |
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| Institution: | 1. Faculty of Biomedical Engineering, Czech Technical University of Prague, Kladno, Czech Republic;2. Institute of Animal Physiology and Genetics, Czech Academy of Science, Libechov, Czech Republic;1. Department of Dental Materials and Prosthesis, School of Dentistry of Ribeirão Preto, University of São Paulo, Ribeirão Preto, Av. do Café, s/n°, 14040-904 Ribeirão Preto, SP, Brazil;2. Department of Biomechanics, Medicine, and Rehabilitation of Locomotive Apparatus, School of Medicine of Ribeirão Preto, University of São Paulo, Ribeirão Preto, Av. Bandeirantes, 3900, 14049-900 Ribeirão Preto, SP, Brazil |
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| Abstract: | 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, Haimovich's 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. |
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