On maximum volume simplices in polytopes
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On maximum volume simplices in polytopes A. V. Akopyan · A. A. Glazyrin
Received: 15 August 2012 / Accepted: 22 August 2013 © Akadémiai Kiadó, Budapest, Hungary 2014
Abstract The paper is devoted to the volume interpretation of Radon’s theorem about partitions. Here we give a new proof of the theorem, show how ideas of this proof can be applied to prisms and parallelotopes, and subsequently generalize several statements by M. Lassak. Keywords
Radon’s theorem · Triangulations · Maximal volume simplex
1 Introduction The theorem published by J. Radon [5] states that any d + 2 point set in Rd can be partitioned into two subsets such that their convex hulls intersect. This partition is unique if points are in general position. This theorem, proved in 1921, is the cornerstone of discrete convex geometry. In particular, it implies Helly’s theorem for convex bodies. As a classical result, Radon’s theorem has many miscellaneous proofs and various generalizations. One of the most famous generalization is the Tverberg theorem [7], which states that for any set of (d + 1)(r − 1) + 1 points in Rd there is a partition into r subsets such that convex hulls of all subsets have a common point. The shortest proof of the Tverberg theorem belongs to K. Sarkaria [6]. He constructed a special (d + 1)(r − 1)-dimensional point set and showed that the Tverberg theorem follows from Barany’s Colorful Caratheodory theorem [1] for this set.
A. V. Akopyan (B) Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia, 127994 e-mail: [email protected] A. V. Akopyan Department of Mathematics, Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia, 141700 A. A. Glazyrin Department of Mathematics, The University of Texas at Brownsville, Brownsville, TX 78520, USA e-mail: [email protected]
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A. V. Akopyan, A. A. Glazyrin
In this article we suggest another point of view on Radon’s theorem that is quite similar to Sarkaria’s approach but we use a volume argument rather than the combinatorial theorem by Barany. Also we use this approach to generalize several statements by M. Lassak [2]. The paper is organized in the following way. In the first section, we formulate a key lemma which is the main tool of the article. In the second section we discuss the Radon theorem and its connection to simplices of maximal volume. In the last section we discuss simplices of maximum volume in prisms and parallelotopes. There we give new proofs of some statements from Lassak’s paper.
2 Key lemma Lemma 2.1 Let B be a centrally symmetric set of 2d + 2 points in Rd whose center is at the origin such that no hyperplane through 0 contains 2d points of B. Let T be a family of simplices with vertices from B without a pair of symmetric points as vertices. Then there are exactly two simplices in T containing the origin in their respective interiors. They are symmetric to each other with respect to the origin and have the maximum Euclidean volume among all simplices in T . Proof Denote the pairs of
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