## Abstract

This paper defines a "connected sum" operation on oriented matroids of the same rank. This construction is used for three different applications in rank 4. First it provides nonrealizable pseudoplane arrangements with a low number of simplicial regions. This contrasts the case of realizable hyperplane arrangements: by a classical theorem of Shannon every arrangement of n projective planes in ℝP^{ d-1} contains at least n simplicial regions and every plane is adjacent to at least d simplicial regions [17], [18]. We construct a class of uniform pseudoarrangements of 4 n pseudoplanes in ℝP^{3} with only 3 n+1 simplicial regions. Furthermore, we construct an arrangement of 20 pseudoplanes where one plane is not adjacent to any simplicial region. Finally we disprove the "strong-map conjecture" of Las Vergnas [1]. We describe an arrangement of 12 pseudoplanes containing two points that cannot be simultaneously contained in an extending hyperplane.

Original language | English |
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Pages (from-to) | 251-269 |

Number of pages | 19 |

Journal | Discrete and Computational Geometry |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1993 |