### Abstract

Let f_{k}(F) denote the number of k-dimensional faces of a d-dimensional arrangement F of spheres or a d-dimensional oriented matroid F. In this paper we show that the following relation among the face numbers is valid: f_{k}(F)≤(^{d}_{k})f_{d}(F) for 0≤k≤d. The same inequalities are valid for d-dimensional arrangements of hyperplanes. Using the result, we obtain a polynomial algorithm to enumerate all faces from the set of maximal faces of an oriented matroid. This algorithm can be applied to any arrangement of hyperplanes in projective space P^{d} or in Euclidean space E^{d}. Combining this with a recent result of Cordovil and Fukuda, we have the following: given the cograph of an arrangement (where the vertices are the d-faces and two vertices are adjacent if they intersect in a (d-1)-face), one can reconstruct the location vectors of all faces of the arrangement up to isomorphism in polynomial time. It is also shown that one can test in polynomial time whether a given set of (+,0,-)-vectors is the set of maximal vectors (topes) of an oriented matroid.

Original language | English |
---|---|

Pages (from-to) | 141-149 |

Number of pages | 9 |

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

## Fingerprint Dive into the research topics of 'Combinatorial face enumeration in arrangements and oriented matroids'. Together they form a unique fingerprint.

## Cite this

*Discrete Applied Mathematics*,

*31*(2), 141-149. https://doi.org/10.1016/0166-218X(91)90066-6