Abstract
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) 2/3 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.
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Katoh, Tokuyama K-Levels of Concave Surfaces. Discrete Comput Geom 27, 567–584 (2002). https://doi.org/10.1007/s00454-001-0086-z
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DOI: https://doi.org/10.1007/s00454-001-0086-z