Computing small hitting sets for convex ranges

Let S be a set of n given points in R 2. If A is a convex subset of R 2, its “size” is defined as| A∩ S|, the number of points of S it contains. We describe an O (n (log n) 4) algorithm to find points z1= z2, at least one of which must meet any convex set of size greater than 4n/7; z1 and z2 comprise a hitting set of size two for such convex ranges. This algorithm can then be used to construct (i) three points, one of which must meet any convex set of size> 8n/15;(ii) four points, one of which must meet any convex set of size> 16n/31;(iii) five points, one of which must meet any convex set of size> 20n/41. Finally, we discuss some open algorithmic and combinatorial problems suggested by these results.Let S be a set of n given points in general position in R 2. If A is a convex subset of R 2, its “size” is defined to be| A∩ S|, the number of points of S that it contains. The (Tukey) depth of a point z∈ R 2 is defined as the minimum (over all halfspaces h containing z) of| S∩ h|, the size of the smallest halfspace containing z. It is familiar that there always exists a point z∈ R 2 (z not necessarily in S) with depth d (z)≥ n/3. Such a point is called a centerpoint for S. The constant c= 1/3 is best-possible: for every c> 1/3 there are sets S with respect to which NO point has depth cn. The interesting algorithm of Jadhav and Mukhopadhyay [2] computes a centerpoint in linear time.