Computational Geometry Column 32

The proof of Dey\u27s new k-set bound is illustrated

Computational Geometry Column 32

The proof of Dey\u27s new k-set bound is illustrated

Computational Geometry Column 32

The proof of Dey's new k-set bound is illustrated. -20-10 10 20 -20 -10 10 20 30 Figure 1: The line shown determines two 5-sets. There are a total of twenty-four 5-sets in this set of n = 10 points. A k-set of a set of n points P in the plane is a set of exactly k points of P contained in an open halfplane. The maximum number of k-sets as a function of n and k ? 0 is an important quantity, but a gap in the known bounds between\Omega\Gamma n log k) and O(nk 1=2 ) stood for over 25 years. The only progress was a tight bound of nk on the number of k-sets, and a slight lowering of the upper bound for k-sets to O(nk 1=2 = log k). 1 But recently, building Department of Computer Science, Smith College, Northampton, MA 01063, USA. [email protected]. Supported by NSF grant CCR-9421670. 1 The first result, which will play a role below, is due to Alon and Gyori [AG86]; the second is due to Pach et al. [PSS92]. See [Ede87] and [AAS97] for history and further references. on the ..