Lines, Circles, Planes and Spheres

Details Lines, Circles, Planes and Spheres Lines, Circles, Planes and Spheres

2009-07-03

arxiv.org/abs/0907.0724v1

Discrete & Computational Geometry, May 2011, Volume 44, Number 4, pp. 860-882

Mathematics

Combinatorics

37 pages

Scientific paper

10.1007/s00454-010-9270-3

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.

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