In this article we consider $S$ to be a set of points in $d$-space with the property that any $d$ points of $S$ span a hyperplane and not all the points of $S$ are contained in a hyperplane. The aim of this article is to introduce the function $e_d(n)$, which denotes the minimal number of hyperplanes meeting $S$ in precisely $d$ points, minimising over all such sets of points $S$ with $|S|=n$.
Ball, S., Monserrat, J. A generalisation of Sylvester's problem to higher dimensions. "Journal of geometry", 1 Juliol 2017, vol. 108, núm. 2, p. 529-543.