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Hyperplane Arrangements with Large Average Diameter Feng Xie with Antoine Deza McMaster University. Line Arrangement A o n ,2 with Maximal Average Diameter. Polytope Diameter. Hyperplane Arrangements A* n ,d with Large Average Diameter. bounded cells of A o :
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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