### 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

Oriented matroid realization (NAKAYAMA code)

Bounded cell enumeration (MINKSUM package)

External facet enumeration (CDD package)

A*6,3

++++++++++

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

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