Interpolation by convex algebraic hypersurfaces
type: Item
instance of: scholarly article
title: Interpolation by convex algebraic hypersurfaces
zbMATH Open document ID: 0872.41002
published in: Journal of Approximation Theory
publication date: 1997-09-11T00:00:00Z
review text: The paper is devoted to the convex algebraic interpolation problem: given a set \(\Sigma\) of vertices of a convex polyhedron in the \(n\)-dimensional Euclidean space, construct a convex algebraic hypersurface of degree as small as possible passing through \(\Sigma\). The main result is that there exists such a hypersurface of degree at most \(d(\Sigma)\), where \(d(\Sigma)\) is the minimal number of simplicial faces of the polyhedron, covering \(\Sigma\). If \(m\) is the number of points in \(\Sigma\), then \(d(\Sigma)\leq(m+1)/2\) in the plane, which implies the Levin-Nadler theorem [\textit{D. Lewin} and \textit{E. Nadler}, Numer. Algorithms 9, No. 1-2, 113-139 (1995; Zbl 0835.41008)], and \(d(\Sigma)<2m/3\) in the three-space. The Hermite-type problem, convex interpolation with given tangent hyperplanes at given points, and the sharpness of upper bounds on \(d(\Sigma)\) are discussed.
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 1006708
zbMATH Keywords: convex interpolation problem
zbMATH Keywords: Hermite-type interpolation
zbMATH Keywords: hyperbolic algebraic hypersurface
author: Eugenii I. Shustin
reviewed by: Eugenii I. Shustin
MaRDI profile type: Publication
full work available at URL: jath.1996.3023
OpenAlex ID: W2072247294
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DOI: 10.1006/JATH.1996.3023
description: scientific article; zbMATH DE number 1006708