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Problem 7: k-sets


What is the maximum number of k-sets? (Equivalently, what is the maximum complexity of a k-level in an arrangement of hyperplanes?)


Uncertain, pending investigation.



Partial and Related Results

For a given set P of n points, S\subset P is a k-set if |S|=k and S=P\cap H for some open halfspace H. Even for points in two dimensions the problem remains open: The maximum number of k-sets as a function of n and k is known to be O(n k^{1/3}) by a recent advance of Dey [Dey98], while the best lower bound is only slightly superlinear [Tot00].




combinatorial geometry; point sets

Entry Revision History

J. O’Rourke, 2 Aug. 2001.



T. K. Dey. Improved bounds on planar k-sets and related problems. Discrete Comput. Geom., 19:373–382, 1998.


J. S. B. Mitchell and Joseph O’Rourke. Computational geometry column 42. Internat. J. Comput. Geom. Appl., 11(5):573–582, 2001. Also in SIGACT News 32(3):63-72 (2001), Issue 120.


Géza Toth. Point sets with many k-sets. In Proc. 16th Annu. ACM Sympos. Comput. Geom., pages 37–42, 2000.