Geometric lower bounds for parametric matroid optimization
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Parametric polymatroid optimization and its geometric applications
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
A Characterization of Planar Graphs by Pseudo-Line Arrangements
ISAAC '97 Proceedings of the 8th International Symposium on Algorithms and Computation
Notes on computing peaks in k-levels and parametric spanning trees
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
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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-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 n2) 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.