Advanced Studies in Pure Mathematics
- Adv. Stud. Pure Math.
- Arrangements of Hyperplanes — Sapporo 2009, H. Terao and S. Yuzvinsky, eds. (Tokyo: Mathematical Society of Japan, 2012), 59 - 74
Hyperplane arrangements with large average diameter: a computational approach
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.
Received: 20 April 2010
Revised: 20 September 2010
First available in Project Euclid: 24 November 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 52C35: Arrangements of points, flats, hyperplanes [See also 32S22]
Secondary: 90C05: Linear programming
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