On sets of points with few ordinary hyperplanes

Let $S$ be a set of $n$ points in the projective $d$-dimensional real space $\mathbb{RP}^d$ such that not all points of $S$ are contained in a single hyperplane and such that any subset of $d$ points of $S$ span a hyperplane. Let an ordinary hyperplane of $S$ be an hyperplane of $\mathbb{RP}^d$ containing exactly $d$ points of $S$. In this paper we study the minimum number of ordinary hyperplanes spanned by any set $S$ of $n$ points in $4$ dimensions, following the work of Ben Green and Terence Tao in the planar version of the problem, as well as the work of Simeon Ball in the $3$ dimensional case. We classify the sets of points in $4$ dimensions that span few ordinary hyperplanes, showing that if $S$ is a set spanning less than $Kn^3$ ordinary hyperplanes, for some $K = o(n^{\frac{1}{6}})$, then all but $O(K)$ points of $S$ must be contained in the intersection of $5$ linearly independent quadrics