On the light side of geometric graphs

Let G be a geometric graph on n vertices in general position in the plane. Suppose that for every line ℓ in the plane the subgraph of G induced by the set of vertices in one of the two half-planes bounded by ℓ has at most k edges ( k ≥ 1 may be a function of n ). Then G has at most O ( n k ) edges. This bound is best possible.