Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Points and triangles in the plane and halving planes in space
Discrete & Computational Geometry
Improved bounds for intersecting triangles and halving planes
Journal of Combinatorial Theory Series A
The colored Tverberg's problem and complexes of injective functions
Journal of Combinatorial Theory Series A
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Lectures on Discrete Geometry
A Characterization of Planar Graphs by Pseudo-Line Arrangements
ISAAC '97 Proceedings of the 8th International Symposium on Algorithms and Computation
On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Lenses in arrangements of pseudo-circles and their applications
Journal of the ACM (JACM)
Intersection reverse sequences and geometric applications
GD'04 Proceedings of the 12th international conference on Graph Drawing
On the bichromatic k-set problem
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
On levels in arrangements of curves, iii: further improvements
Proceedings of the twenty-fourth annual symposium on Computational geometry
On the bichromatic k-set problem
ACM Transactions on Algorithms (TALG)
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A favorite open problem in combinatorial geometry is to determine the worst-case complexity of a level in an arrangement. Up to now, nontrivial upper bounds in three dimensions are known only for the linear cases of planes and triangles. We propose the first technique that can deal with more general surfaces in three dimensions. For example, in an arrangement of n "pseudo-planes" or "pseudo-spheres" (where each triple of surfaces has at most two common intersections), we prove that there are at most O(n2.9986) vertices of any given level.