Error of asymptotic formulae for volume approximation of convex bodies in \({\mathbb E}^d\)
type: Item
instance of: scholarly article
title: Error of asymptotic formulae for volume approximation of convex bodies in \({\mathbb E}^d\)
zbMATH Open document ID: 1006.52002
published in: Monatshefte für Mathematik
publication date: 2002-07-29T00:00:00Z
review text: In the paper under review the error of asymptotic formulae for volume approximation of a sufficiently smooth convex body \(C\subset {\mathbb R}^d\) by polytopes \(P_n\) is estimated as the number \(n\) of facets of \(P_n\) tends to infinity. The author [Forum Math. 5, No. 6, 521-537 (1993; Zbl 0788.41020)] derived a precise asymptotic formula for the volume of the symmetric difference, \(\delta(C,P_n)\), of \(C\) and its best approximating polytopes \(P_n\). The order of approximation is \(\displaystyle a(C)/ n^{2/(d-1)}\) where \(a(C)>0\) depends on the affine surface area of \(C\). Here the author refines his result and shows that the error term \[ \delta(C,P_n)-a(C)/n^{2/(d-1)}=O(\displaystyle 1/ n^{2/(d-1) + 1/(3 (d-1))-\varepsilon}) \] as \(n\to\infty\) for every \(\varepsilon>0\). He conjectures that the precise order of this error term is \(a'(C)/n^{3/(d-1)}\). The result is applied to estimate the minimum isoperimetric quotient of polytopes \(P_n\) in Minkowski geometry as \(n\to\infty\).
reviewed by: Monika Ludwig
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 52A27
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 1773410
zbMATH Keywords: volume approximation
zbMATH Keywords: asymptotic formulae
zbMATH Keywords: isoperimetric problems
zbMATH Keywords: Minkowski geometry
author: Peter M. Gruber
MaRDI profile type: Publication
full work available at URL: s006050200023
OpenAlex ID: W2058101030
DOI: 10.1007/S006050200023
description: scientific article; zbMATH DE number 1773410