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- W3104636610 abstract "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." @default.
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- W3104636610 date "2010-06-04" @default.
- W3104636610 modified "2023-09-27" @default.
- W3104636610 title "Lines, Circles, Planes and Spheres" @default.
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- W3104636610 doi "https://doi.org/10.1007/s00454-010-9270-3" @default.
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