Bounding the number of k-faces in arrangements of hyperplanes

Komei Fukuda, Shigemasa Saito, Akihisa Tamura, Takeshi Tokuyama

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


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
Issue number2
Publication statusPublished - 1991 Apr 15
Externally publishedYes

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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