### 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=(f_{0},f_{1},...,f_{d}) of an arrangement, where f_{k} 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 f_{k}>(d-k+1) kf_{--1} if f_{k}≠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 language | English |
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Pages (from-to) | 151-165 |

Number of pages | 15 |

Journal | Discrete Applied Mathematics |

Volume | 31 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1991 Apr 15 |

Externally published | Yes |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Discrete Applied Mathematics*,

*31*(2), 151-165. https://doi.org/10.1016/0166-218X(91)90067-7