We show that for any 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 Lovász's lemma, which is a key constituent in the analysis of the complexity of k-levels of planes. Our proof is constructive, and finds a family of concave surfaces covering the "laminated at-most-k-level." As a consequence, (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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics