Solidity indices for convex cones
type: Item
instance of: scholarly article
title: Solidity indices for convex cones
zbMATH Open document ID: 1276.28011
author: Daniel Gourion
author: Alberto Seeger
published in: Positivity
publication date: 2013-01-02T00:00:00Z
review text: A solidity index \(\sigma\) is a continuous real function defined on the set \(\Xi_n\) of non-trivial closed convex cones of \({\mathbb R}^n\), which is non-decreasing (with respect to set inclusion), invariant under orthogonal transformations, and so that it vanishes if the cone \(K\) is flat, whereas its value is \(1\) if \(K\) is a half-space. In this paper, the authors consider three particularly interesting solidity indices: the metric solidity index \(\sigma_{\operatorname{met}}(K)=\min_{P\in\Xi_n,\,P\mathrm{ is flat}} \delta(K\cap B_n,P\cap B_n)\) (here \(\delta\) is the Hausdorff distance and \(B_n\) the Euclidean unit ball); the volumetric solidity index \(\sigma_{\operatorname{vol}}(K)=2\operatorname{vol}(K\cap B_n)/\operatorname{vol}(B_n)\) (\(\operatorname{vol}\) denotes the \(n\)-dimensional volume); and the Frobenius solidity index \(\sigma_{\operatorname{frob}}(K) \) is the radius of the largest ball centered at a unit vector and contained in \(K\). They study properties of these indices (Lipschitz behavior of \(\sigma_{\operatorname{vol}}\) or numerical computation of \(\sigma_{\operatorname{frob}}\)) and establish all the relations between those three functions.
reviewed by: Maria A. Hernández Cifre
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 6121306
zbMATH Keywords: convex cone
zbMATH Keywords: solidity index
MaRDI profile type: Publication
full work available at URL: s11117-011-0140-x
OpenAlex ID: W1995892131
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DOI: 10.1007/S11117-011-0140-X
description: scientific article; zbMATH DE number 6121306