Advanced Studies in Pure Mathematics

Hyperplane arrangements with large average diameter: a computational approach

Antoine Deza, Hiroyuki Miyata, Sonoko Moriyama, and Feng Xie

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Abstract

We consider the average diameter of a bounded cell of a simple arrangement defined by $n$ hyperplanes in dimension $d$. In particular, we investigate the conjecture stating that the average diameter is no more than the dimension $d$. Previous results in dimensions 2 and 3 suggested that specific extensions of the cyclic arrangement might achieve the largest average diameter. We show that the suggested arrangements do not always achieve the largest diameter and disprove a related conjecture dealing with the minimum number of facets belonging to exactly one bounded cell. In addition, we computationally determine the largest possible average diameter in dimensions 3 and 4 for arrangements defined by no more than 8 hyperplanes via the associated uniform oriented matroids. These new entries substantiate the hypothesis that the largest average diameter is achieved by an arrangement minimizing the number of facets belonging to exactly one bounded cell. The computational framework to generate specific arrangements, and to compute the average diameter and the number of facets belonging to exactly one bounded cell is presented.

Article information

Source
Arrangements of Hyperplanes — Sapporo 2009, H. Terao and S. Yuzvinsky, eds. (Tokyo: Mathematical Society of Japan, 2012), 59-74

Dates
Received: 20 April 2010
Revised: 20 September 2010
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543085004

Digital Object Identifier
doi:10.2969/aspm/06210059

Mathematical Reviews number (MathSciNet)
MR2933792

Zentralblatt MATH identifier
1261.52014

Subjects
Primary: 52C35: Arrangements of points, flats, hyperplanes [See also 32S22]
Secondary: 90C05: Linear programming

Keywords
Simple hyperplane arrangements average diameter Hirsch conjecture oriented matroids

Citation

Deza, Antoine; Miyata, Hiroyuki; Moriyama, Sonoko; Xie, Feng. Hyperplane arrangements with large average diameter: a computational approach. Arrangements of Hyperplanes — Sapporo 2009, 59--74, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06210059. https://projecteuclid.org/euclid.aspm/1543085004


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