### 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 |

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### ASJC Scopus subject areas

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

### Cite this

*Discrete Applied Mathematics*,

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

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

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 31, no. 2, pp. 151-165. https://doi.org/10.1016/0166-218X(91)90067-7

}

TY - JOUR

T1 - Bounding the number of k-faces in arrangements of hyperplanes

AU - Fukuda, Komei

AU - Saito, Shigemasa

AU - Tamura, Akihisa

AU - Tokuyama, Takeshi

PY - 1991/4/15

Y1 - 1991/4/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0004515166&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0004515166&partnerID=8YFLogxK

U2 - 10.1016/0166-218X(91)90067-7

DO - 10.1016/0166-218X(91)90067-7

M3 - Article

VL - 31

SP - 151

EP - 165

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 2

ER -