The number of small semispaces of a finite set of points in the plane

For a configuration S of n points in the plane, let gk(S) denote the number of subsets of cardinality <k cut off by a line. Let gk, " =max{gk(S): (SI =n}. Good-man and Pollack (J. Combin. Theory Ser. A 36 (1984), 101-104) showed that if k < n/2 then g,,, < 2nk-2k2-k. Here we show that g,, " = k. n for k <n/2. Let S be a finite set of points in the plane. Following Goodman and Pollack [GP2] we call the intersection of S with a half plane a semispace of S. A semispace of S of cardinality k is called a k-set of S. Let f,JS) denote the number of k-sets of S and put gk(S) = Cr= I f,(S). Define Thus gk,n is the maximal number of (6 k)-sets of n points in the plane. Since gk,n = &-k,n we may restrict our attention to the case k 6 n/2. Goodman and Pollack [GP2] considered the problem of estimating g,., and proved that if k < n/2 then g,., < 2nk-2k2-k.