Lines, Circles, Planes and Spheres

Let \(S\) be a set of \(n\) points in \(\mathbb{R}^3\), no three collinear and not all coplanar. If at most \(n-k\) are coplanar and \(n\) is sufficiently large, the total number of planes determined is at least \(1 + k \binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2})\). For similar conditions and sufficiently large \(n\), (inspired by the work of P. D. T. A. Elliott in \cite{Ell67}) we also show that the number of spheres determined by \(n\) points is at least \(1+\binom{n-1}{3}-t_3^{orchard}(n-1)\), and this bound is best possible under its hypothesis. (By \(t_3^{orchard}(n)\), we are denoting the maximum number of three-point lines attainable by a configuration of \(n\) points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles.