More onk-sets of finite sets in the plane

For a setS ofn points in the plane and forK {1, 2, ..., 1/2n]}, letf _K (S) denote the number of subsets ofS with cardinalityk K which can be cut offS by a straight line. We show that there is a positive constantc such thatf _K (S)<cn ( _{k K} k)^1/2.This research was carried out during the author's stay at the University of Leiden, The Netherlands.