### Abstract

We consider sets and maps defined over an o-minimal structure over the reals, such as real semi-algebraic or globally subanalytic sets. A monotone map is a multi-dimensional generalization of a usual univariate monotone continuous function on an open interval, while the closure of the graph of a monotone map is a generalization of a compact convex set. In a particular case of an identically constant function, such a graph is called a semi-monotone set. Graphs of monotone maps are, generally, non-convex, and their intersections, unlike intersections of convex sets, can be topologically complicated. In particular, such an intersection is not necessarily the graph of a monotone map. Nevertheless, we prove a Helly-type theorem, which says that for a finite family of subsets of {Mathematical expression}, if all intersections of subfamilies, with cardinalities at most {Mathematical expression}, are non-empty and graphs of monotone maps, then the intersection of the whole family is non-empty and the graph of a monotone map.

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

Pages (from-to) | 857-864 |

Number of pages | 8 |

Journal | Discrete & Computational Geometry |

Volume | 50 |

Issue number | 4 |

Early online date | 17 Sep 2013 |

DOIs | |

Publication status | Published - Dec 2013 |

### Keywords

- Monotone maps, Helly's theorem

## Fingerprint Dive into the research topics of 'A Helly-type theorem for semi-monotone sets and monotone maps'. Together they form a unique fingerprint.

## Cite this

Basu, S., Gabrielov, A., & Vorobjov, N. (2013). A Helly-type theorem for semi-monotone sets and monotone maps.

*Discrete & Computational Geometry*,*50*(4), 857-864. https://doi.org/10.1007/s00454-013-9540-y