Partitions of points into intersecting tetrahedra
Radon's theorem asserts that any set S of \(d+2\) point in \({\mathbb{R}}^ d\) has a partition into two subsets \(S_ 1\) and \(S_ 2\) such that \(Conv(S_ 1)Conv(S_ 2)\neq \emptyset\) where \(Conv(S_ i)\) denotes the convex hull of \(S_ i\) Tverberg proved its generalization stated by the following theorem: Any set S of \((m-1)(d+1)+1\) points in \({\mathbb{R}}^ d\) has a partition into m disjoint subsets \(S_ 1,S_ 2,...,S_ m\) such that \(Conv(S_ 1)\cap Conv(S_ 2)\neq \emptyset,\) where \(Conv(S_ i)\) denotes the convex hull of \(S_ i\). Tverberg proved its generalization stated by the following theorem: Any set S of \((m-1)(d+1)\) points in \({\mathbb{R}}^ d\) has a partition into m disjoint subsets \(S_ 1,S_ 2,...,S_ m\) such that \(\cap^{m}_{i=1}Conv(S_ i)\neq \emptyset.\) Reay conjectured that any set of (m-1) \((d+1)+k+1\) points in general position in \({\mathbb{R}}^ d\), with \(0\leq k\leq d\) has a partition into m disjoint subsets \(S_ 1,S_ 2,...,S_ m\) such that \(\cap^{m}_{i=1}Conv(S_ i)\) is k-dimensional. The author proves this conjecture for \(d=3\).
scientific article; zbMATH DE number 4142045