Lovasz's lemma for the three-dimensional k-level of concave surfaces and its applications

Naoki Katoh, Takeshi Tokuyama

Research output: Contribution to journalConference articlepeer-review

4 Citations (Scopus)


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)2/3n2) 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 languageEnglish
Pages (from-to)389-398
Number of pages10
JournalAnnual Symposium on Foundations of Computer Science - Proceedings
Publication statusPublished - 1999 Dec 1
EventProceedings of the 1999 IEEE 40th Annual Conference on Foundations of Computer Science - New York, NY, USA
Duration: 1999 Oct 171999 Oct 19

ASJC Scopus subject areas

  • Hardware and Architecture

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