Hyperplane arrangements with large average diameter feng xie with antoine deza mcmaster university
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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 Arrangements with Large Average Diameter Feng Xie with Antoine Deza McMaster University

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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


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