Convex sets with large distortion
type: Item
instance of: scholarly article
title: Convex sets with large distortion
zbMATH Open document ID: 1172.52003
author: Gábor Tóth
published in: Journal of Geometry
publication date: 2009-06-08T00:00:00Z
review text: Let \(C\in\partial\mathcal L\) be a boundary point of a compact convex body \(\mathcal L\) in an \(m\)-dimensional Euclidean vector space with a fixed base point \(\mathcal O\) in int\(\mathcal L\). With the line \(l({\mathcal O}C)\) and \(C^o:=l({\mathcal O}C)\cap \partial{\mathcal L}\setminus \{C\}\) the distortion function \(\Lambda\) is defined by \(\Lambda(C):= d(C,{\mathcal O}) / d(C^o,\mathcal O)\)where \(d\) denotes the Euclidean distance function. For appropriate \(\mathcal O\) this function satisfies \(1/m \leq \Lambda \leq m\). With the aid of \(\Lambda\) an affine invariant \(\sigma (\mathcal L)\) can be defined that measures how distorted \(\mathcal L\) is with respect to \(\mathcal O\). The author continues his earlier investigations of convex bodies [see, for example, Trans. Am. Math. Soc. 358, No. 6, 2425--2446 (2006; Zbl 1108.53032)]. In the present paper he proves that if \(\sigma ({\mathcal L}) < 1+1/(m+1)\) then \(\mathcal L\) is between two special homothetic simplices. That gives an upper bound on the volume of \(\mathcal L\) in terms of \(\sigma(\mathcal L)\) and the inscribed simplex.
reviewed by: Eike Hertel
Mathematics Subject Classification ID: 52A05
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 5562642
zbMATH Keywords: convex bodies
zbMATH Keywords: distortion
MaRDI profile type: Publication
full work available at URL: s00022-009-1901-6
OpenAlex ID: W1994145846
DOI: 10.1007/S00022-009-1901-6
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description: scientific article; zbMATH DE number 5562642