Hyperplane Arrangements with Large Average Diameter Feng Xie with Antoine Deza McMaster University

1. Hyperplane Arrangementswith Large Average DiameterFeng Xie with Antoine DezaMcMaster University Line Arrangement Aon,2 with Maximal Average Diameter Polytope Diameter Hyperplane Arrangements A*n,dwith Large Average Diameter • bounded cells ofAo: 4-gons n - 2 triangles 1 n-gon • (Ao)=2 - • bounded cells of A*: • cubical cells • (n - d)(n – d - 1) simplex prisms • n-d simplices • (A*) ≥ n = 7 d= 2 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 Computational Framework Ao minimizes external facets (2n – 2) and maximizes average diameter • Enumeration of Arrangements Finschi’s online database of oriented matroids (www.om.math.ethz.ch) Average Diameter of an Arrangement Plane Arrangements A*n,3 & Aon,3 with Large Average Diameter • Algorithm Overview Oriented matroid realization (NAKAYAMA code) Bounded cell enumeration (MINKSUM package) External facet enumeration (CDD package) A*6,3 • bounded cells ofA*: cubical cells (n - 3)(n - 4) triangular prisms n - 3 tetrahedra ++++++++++ i (A) : average diameter of a bounded cell of A: (A) = with I = • A* mainly consists of cubical cells ii iii (showing only cells in positive orthant) Ao6,3 Conjecture (Deza, Terlaky and Zinchenko):(A) ≤ d • It is the discrete analogue of Dedieu-Malajovich-Shub 2005result: the average curvature of the central path is less than 2 d • bounded cells ofAo: cubical cells (n - 3)(n - 4)-1 triangular prisms n - 3 tetrahedra 1 n-shell 7-shell Future works • maximal (An,3) = ? • ultilize oriented matroid & algebrato study (An,d) • Indication on Hirsch conjecture ? • Incorporate rational or high-precision computation • Aon,3 does not maximize average diameter • Hirsch conjecture implies (A) ≤ d