### Abstract

We show that for line l in space, there are at most k(k + 1) tangent planes through l to the k-level of an arrangement of concave surfaces. This is a generalization of Lovasz's lemma, which is a key constituent in the analysis of the complexity of k-level of planes. Our proof is constructive, and finds a family of concave surfaces covering the 'laminated at-most-k level'. As consequences, (1): we have an O((n-k)^{2/3}n^{2}) upper bound for the complexity of the k-level of n triangles of space, and (2): we can extend the k-set result in space to the k-set of a system of subsets of n points.

Original language | English |
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Pages (from-to) | 389-398 |

Number of pages | 10 |

Journal | Annual Symposium on Foundations of Computer Science - Proceedings |

Publication status | Published - 1999 Dec 1 |

Event | Proceedings of the 1999 IEEE 40th Annual Conference on Foundations of Computer Science - New York, NY, USA Duration: 1999 Oct 17 → 1999 Oct 19 |

### ASJC Scopus subject areas

- Hardware and Architecture

## Cite this

Katoh, N., & Tokuyama, T. (1999). Lovasz's lemma for the three-dimensional k-level of concave surfaces and its applications.

*Annual Symposium on Foundations of Computer Science - Proceedings*, 389-398.