On the number of k-subsets of a set of n points in the plane

For a configuration S of n points in $\text{E}$², H. Edelsbrunner (personal communication) has asked for bounds on the maximum number of subsets of size k cut off by a line. By generalizing to a combinatorial problem, we show that for 2k < n the number of such sets of size at most k is at most 2nk ? 2k² ? k. By duality, the same bound applies to the number of cells at distance at most k from a base cell in the cell complex determined by an arrangement of n lines in $\text{P}$².