Bounding the number of k-faces in arrangements of hyperplanes

Komei Fukuda, Shigemasa Saito, Akihisa Tamura, Takeshi Tokuyama

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

We study certain structural problems of arrangements of hyperplanes in d-dimensional Euclidean space. Of special interest are nontrivial relations satisfied by the f-vector f=(f0,f1,...,fd) of an arrangement, where fk denotes the number of k-faces. The first result is that the mean number of (k-1)-faces lying on the boundary of a fixed k-face is less than 2k in any arrangement, which implies the simple linear inequality fk>(d-k+1) kf--1 if fk≠0. Similar results hold for spherical arrangements and oriented matroids. We also show that the f-vector and the h-vector of a simple arrangement is logarithmic concave, and hence unimodal.

Original languageEnglish
Pages (from-to)151-165
Number of pages15
JournalDiscrete Applied Mathematics
Volume31
Issue number2
DOIs
Publication statusPublished - 1991 Apr 15
Externally publishedYes

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Arrangement of Hyperplanes
Arrangement
Face
F-vector
H-vector
Oriented Matroid
Euclidean space
Linear Inequalities
Logarithmic
Denote
Imply

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Applied Mathematics
  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Bounding the number of k-faces in arrangements of hyperplanes. / Fukuda, Komei; Saito, Shigemasa; Tamura, Akihisa; Tokuyama, Takeshi.

In: Discrete Applied Mathematics, Vol. 31, No. 2, 15.04.1991, p. 151-165.

Research output: Contribution to journalArticle

Fukuda, Komei ; Saito, Shigemasa ; Tamura, Akihisa ; Tokuyama, Takeshi. / Bounding the number of k-faces in arrangements of hyperplanes. In: Discrete Applied Mathematics. 1991 ; Vol. 31, No. 2. pp. 151-165.
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