Radii of simplices and some applications to geometric inequalities
type: Item
instance of: scholarly article
title: Radii of simplices and some applications to geometric inequalities
zbMATH Open document ID: 1077.52009
author: René Brandenberg
author: Thorsten Theobald
published in: Beiträge zur Algebra und Geometrie
publication date: 2004-09-10T00:00:00Z
full work available at URL: https://eudml.org/doc/228896
full work available at URL: 19.html
review text: In any Euclidean space \({\mathbf E}^d\), bodies have circumradii and inradii (though not necessarily unique inspheres). Moreover, for dimensions \(1\leq j\leq d\), we can define the outer \(j\)-radius \(R_j(B)\) of a body \(B\) as the smallest radius such that some \(j\)-ball of that radius contains the projection of \(B\) into some \(j\)-dimensional subspace, and the inner \(j\)-radius \(r_j(B)\) as the radius of the largest \(j\)-ball contained in \(B\). This paper is largely a (very readable) review of the literature on inequalities between these various radii. Some new results are given, notable a counterexample showing that a certain theorem of Gritzmann and Klee about circumradii does not generalize to outer \(j\)-radii with \(j<d\).
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 2100245
zbMATH Keywords: Radii
zbMATH Keywords: geometric inequalities
zbMATH Keywords: simplices
zbMATH Keywords: convex bodies
zbMATH Keywords: enclosing cylinders
zbMATH Keywords: orthogonal projections
reviewed by: Robert Dawson
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2100245