The Number of k-Point Subsets Separable by Convex Pseudo-Circles

We show that any set S of n points in the plane contains 2kn−n−k 2 +1− Pk−1 i=1 a (i) (S) subsets of k points that can be separated from the rest of S by convex pseudo-circles (where a (i) (S) denotes the number of i-sets of S). This value does not depend on the set of pseudo-circles.