On the number of directions determined by a three-dimensional points set

Let P be a set of n points in R3, not all of which are in a plane and no three on a line. We partially answer a question of Scott (1970) by showing that the connecting lines of P assume at least 2n− 3 different directions if n is even and at least 2n − 2 if n is odd. These bounds are sharp. The proof is based on a far-reaching generalization of Ungar’s theorem concerning the analogous problem in the plane.