Beck's theorem is a classical result in discrete geometry which describes about the geometry of points in the plane. The result states that a finite collections of points in the plane fall into one of two extremes; one where a large fraction of points lie on a single line, and one where a large number of lines are needed to connect all the points.
Formally, the statement of the theorem is as follows:
Theorem: Let $S$ be a set of $n$ points in the plane. If at most $(n-k)$ points lie on any line for some $0 \leq k < n-2$, then there exist $\Omega(nk)$ lines determined by the points of $S$.
Now, my question is that if the generalization/variation of the above result is known in the case of higher dimension $(dim\geq 3)$?