On the approximation of convex bodies by convex algebraic level surfaces
type: Item
instance of: scholarly article
title: On the approximation of convex bodies by convex algebraic level surfaces
zbMATH Open document ID: 1194.52005
author: András Kroó
published in: Journal of Approximation Theory
publication date: 2010-05-11T00:00:00Z
review text: \textit{H. Minkowski} [Math. Ann. 57, 447--495 (1903; JFM 34.0649.01)] showed that the boundary of any convex body in \(\mathbb{R}^d\), can be approximated arbitrarily well by a level surface of a convex analytic function. \textit{P. C. Hammer} [Mathematika 10, 64--71 (1963; Zbl 0122.41004)] generalized Minkowski's result by showing that his approximation result remains valid for approximation by convex algebraic level surfaces. In [\textit{A. Kroó}, Jaen J. Approx. 1, No.~1, 97--109 (2009; Zbl 1183.41010)] Hammer's approximation theorem was quantified by showing that the order of approximation of convex surfaces by convex algebraic level surfaces of degree \(n\) is bounded from above by \({c\log n\over n}\) and it was also conjectured that the \(\log n\) term in this upper bound can be omitted. In this paper, the author shows that the boundary of every convex body in \(\mathbb{R}^d\) can be approximated by level surfaces of convex algebraic polynomials of degree at most \(n\) with order \(1/n\). It is also shown that in a certain sense this order of approximations is the best possible.
Mathematics Subject Classification ID: 52A27
zbMATH DE Number: 5706953
zbMATH Keywords: convex body
zbMATH Keywords: analytic function
zbMATH Keywords: convex algebraic level surface
reviewed by: T. D. Narang
MaRDI profile type: Publication
full work available at URL: j.jat.2009.08.005
OpenAlex ID: W2139782581
cites work: Approximation of convex surfaces by algebraic surfaces
cites work: Q3633870
cites work: Rational approximation to \(|x|\)
cites work: Q5667036
cites work: Q5791308
DOI: 10.1016/J.JAT.2009.08.005
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description: scientific article; zbMATH DE number 5706953