On the Number of Line Separations of a Finite Set in the Plane

Abstract Let S denote a set of n points in the Euclidean plane. A subset S ′ of S is termed a k -set of S if it contains k points and there exists a straight line which has no point of S on it and separates S ′ from S − S ′. We let f k ( n ) denote the maximum number of k -sets which can be realized by a set of n points. This paper studies the asymptotic behaviour of f k ( n ) as this function has applications to a number of problems in computational geometry. A lower and an upper bound on f k ( n ) is established. Both are nontrivial and improve bounds known before. In particular, f k (n) = f n−k (n) = Ω(n log k) is shown by exhibiting special point-sets which realize that many k -sets. In addition, f k (n) = f n−k (n) = O(nk 1 2 ) is proved by the study of a combinatorial problem which is of interest in its own right.