On maximum volume simplices in polytopes
The authors give an alternative proof of Radon's theorem that any set of \(d + 2\) points in general position in \(\mathbb{R}^d\) has a unique partition into two disjoint sets whose convex hulls intersect. In their method, a partition of the \(d+2\) points is associated with a \((d+1)\)-simplex in \(\mathbb{R}^{d+1}\). A volume argument and the following key lemma then yield the desired result. If \(B\) is a centrally symmetric set of \(2d + 4\) points in \(\mathbb{R}^{d+1}\) with center the origin such that no \(2d+2\) points of \(B\) are on a hyperplane through 0, then exactly two simplices in the family \(\mathcal{T}\) of all \((d+1)\)-dimensional simplices with vertices from \(B\) without a pair of symmetric points as vertices contain the origin in their respective interiors. These simplices are symmetric to each other with respect to the origin and are also characterized by having maximum Euclidean volume among all simplices in \(\mathcal{T}\). The authors further use their method to obtain alternative proofs of some results of \textit{M. Lassak} [Discret. Comput. Geom. 21, No. 3, 449--462 (1999; Zbl 0927.52019)] about \(d\)-dimensional simplices of maximum volume inscribed in a \(d\)-dimensional parallelotope.
scientific article; zbMATH DE number 6528020