Polytopes and arrangements: Diameter and curvature
Section snippets
Continuous analogue of the conjecture of Hirsch
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 a simple arrangement is less than the dimension. We prove a continuous analogue of the result of Holt–Klee, namely, we construct a family of polytopes which attain the conjectured
Discrete analogue of the result of Dedieu, Malajovich, Shub
Let be a simple arrangement formed by m hyperplanes in dimension n. We recall that an arrangement is called simple if and any n hyperplanes intersect at a unique distinct point. Since is simple, the number of bounded cells, i.e., bounded connected components of the complement to the hyperplanes, of is . Let denote the total curvature of the primal central path corresponding to . Following the approach of Dedieu et al. [4], let denote the average value
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Proposition 3.1 If the conjecture of Hirsch holds, then . Proof Let denote the number of hyperplanes of which are non-redundant for the description of a bounded cell . If the conjecture of Hirsch holds, we have . It implies Since a facet belongs to at most 2 cells, the sum of for is less than twice the number of bounded facets of . As a bounded facet induced by a hyperplane H of corresponds to a bounded cell of the -dimensional simple
Acknowledgments
Research supported by an NSERC Discovery grant, by a MITACS grant and by the Canada Research Chair program.
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Cited by (16)
Diameter and Curvature: Intriguing Analogies
2008, Electronic Notes in Discrete MathematicsThe tropical analogue of the Helton–Nie conjecture is true
2019, Journal of Symbolic ComputationCitation Excerpt :For instance, Itenberg and Viro (1996) disproved the Ragsdale conjecture as an application of the tropical patchworking method. More recently, Allamigeon et al. (2018a) contradicted, by a tropical method, the continuous analogue of the Hirsch conjecture proposed by Deza et al. (2008). The validity of the tropical analogue of the Helton–Nie conjecture raises the question whether a counterexample could be found by a tropical approach, for instance, by studying images of convex semialgebraic sets and spectrahedra through a map carrying more information than the valuation.
New redundant constraints for Klee-Minty problem
2021, International Journal of Mathematical Modelling and Numerical Optimisation