Problem 7: k-sets

Statement

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

Origin

Uncertain, pending investigation.

Status/Conjectures

Open.

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(nk^1/3) by a recent advance of Dey [Dey98], while the best lower bound is only slightly superlinear [Tot00].

Appearances

[MO01]

Categories

combinatorial geometry; point sets

Entry Revision History

J. O'Rourke, 2 Aug. 2001.

Bibliography

Dey98

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

MO01

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.

Tot00

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