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# Hyperplane Arrangements with Large Average Diameter Feng Xie with Antoine Deza McMaster University - PowerPoint PPT Presentation

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