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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 arranement 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 nds a family of concave surfaces covering the "laminated at-most-k level" . As consequences, (1): we have an O((n k) 2=3 n 2 ) upper bound for the complexity of the k-level of n triangles ... (Update)
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BibTeX entry: (Update)
@misc{ katoh-lovszs,
author = "Naoki Katoh and Takeshi Tokuyama",
title = "Lovász's Lemma for the Three-Dimensional K-Level of Concave Surfaces and
its Applications",
url = "citeseer.ist.psu.edu/katoh99lovszs.html" }
Citations (may not include all citations):42 Improved Bound on Planar k-Sets and Related Problems (context) - Dey - 1998
31 Counting Triangle Crossings and Halving Planes (context) - Dey, Edelsbrunner - 1994
30 On Levels in Arrangement of Lines (context) - Agarwal, Aronov et al. - 1998
29 Annals of Discrete Mathematics (context) - Fujishige, Functions - 1991
22 the Number of Halving Planes (context) - ar, uredi et al. - 1990
21 How to Cut Pseudo-Parabolas into Segments (context) - Tamaki, Tokuyama - 1998
18 Convexity and Steinitz's Exchange Property (context) - Murota - 1996
18 the Number of Halving Lines (context) - Lov - 1971
15 Geometric Lower Bounds for Parametric Matroid Optimization - Eppstein - 1995
14 On Minimum and Maximum Spanning Trees of Linearly Moving Poi.. (context) - Katoh, Iwano et al. - 1995
11 A Characterization of Planar Graphs by Pseudo-line Arrangeme.. (context) - Tamaki, Tokuyama - 1997
10 A Bound on Local Minima of Arrangement That Implies the Uppe.. - Clarkson - 1993
3 Parametric Polymatroid Optimization and Its Geometric Applic.. (context) - Katoh, Tamaki et al. - 1999
3 Polytopes in Arrangements - Aronov, Dey - 1999
1 Bound for the Parametric Spanning Tree Problem (context) - Guseld - 1979
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