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

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.