On maximum volume simplices in polytopes
type: Item
instance of: scholarly article
title: On maximum volume simplices in polytopes
zbMATH Open document ID: 1340.52005
published in: Periodica Mathematica Hungarica
publication date: 2016-01-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 6528020
zbMATH Keywords: Radon's theorem
zbMATH Keywords: triangulations
zbMATH Keywords: maximal volume simplex
zbMATH Keywords: parallelotope
zbMATH Keywords: convex hull
author: Arseniy Akopyan
author: Alexey Glazyrin
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
full work available at URL: s10998-014-0051-2
OpenAlex ID: W2049341446
cites work: A generalization of Caratheodory's theorem
cites work: Parallelotopes of maximum volume in a simplex
cites work: On a property of \(n\)-dimensional simplices
cites work: Properties of axial diameters of a simplex
cites work: Tverberg's theorem via number fields
cites work: A Generalization of Radon's Theorem
DOI: 10.1007/S10998-014-0051-2
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description: scientific article; zbMATH DE number 6528020