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Hyperplane Arrangements with Large Average Diameter Feng Xie with Antoine Deza McMaster UniversityPowerPoint Presentation

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

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

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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