Hyperplane Arrangements with Large Average Diameter Feng Xie
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Hyperplane Arrangements with Large Average Diameter Feng Xie with Antoine Deza Mc. Master University Polytope Diameter o A Line Arrangement n, 2 with Maximal Average Diameter Hyperplane Arrangements A*n, d with Large Average Diameter q bounded cells of Ao : q bounded cells of A*: n =7 d =2 cubical cells 4 -gons n-2 triangles 1 (n - d)(n – d - 1) n-d simplex prisms simplices n-gon q (A*) ≥ Diameter (P): smallest number such that any two vertices can be connected by a path with at most (P) edges Hirsch Conjecture (1957): (P) ≤ n - d Average Diameter of an Arrangement q (Ao) = 2 - Computational Framework Ao minimizes external facets (2 n – 2) and maximizes average diameter o A Plane Arrangements A*n, 3 & n, 3 with Large Average Diameter A*6, 3 q bounded cells of A* : cubical cells (n - 3)(n - 4) n-3 (A) : average diameter of a bounded cell of A: (A) = Finschi’s online database of oriented matroids (www. om. math. ethz. ch) q Algorithm Overview i. Oriented matroid realization (NAKAYAMA code) ii. Bounded cell enumeration (MINKSUM package) iii. External facet enumeration (CDD package) triangular prisms tetrahedra q A* mainly consists of cubical cells +++++ i ii with I = Ao Conjecture (Deza, Terlaky and Zinchenko) : (A) ≤ d q Enumeration of Arrangements 6, 3 (showing only cells in positive orthant) iii q bounded cells of Ao : cubical cells q It is the discrete analogue of Dedieu. Malajovich-Shub 2005 result: 7 -shell (n - 3)(n - 4)-1 triangular prisms n-3 the average curvature of the central path is less than 2 d q Hirsch conjecture implies (A) ≤ d 1 tetrahedra n-shell q Aon, 3 does not maximize average diameter Future works • maximal (An, 3) = ? • ultilize oriented matroid & algebra to study (An, d) • Indication on Hirsch conjecture ? • Incorporate rational or high-precision computation