The number of extreme pairs of finite point-sets in Euclidean spaces (Q1087828)

For a set S of n points in \(E^ d\) let \(e^ k_ k(n)\) denote the maximum number of k-sets S' such that \(S'=S\cap h\), where h is a half space. Theorem. (i) \(e^ 2_ 2=\lfloor 3n/2\rfloor\) for \(n\geq 4\), (ii) \(e^ 3_ 2=10\) and \(e^ 3_ 2(n)=3n-6\) for \(n\geq 6\), (iii) \(e^ d_ 2(n)=\left( \begin{matrix} n\\ 2\end{matrix} \right)\) for \(4\leq d\leq n-2\).