Improved bounds for planar k-sets and related problems

Abstract

We prove an O(n(k + 1)1/3) upper bound for planar k-sets. This is the first considerable improvement on this bound after its early solution approximately 27 years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of k-levels in the arrangement of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees, and parametric matroids in general.