The length of a set in the sphere whose polynomial hull contains the origin
type: Item
instance of: scholarly article
title: The length of a set in the sphere whose polynomial hull contains the origin
zbMATH Open document ID: 0762.32007
DOI: 10.1016/0019-3577(92)90004-5
author: Franc Forstnerič
published in: Indagationes Mathematicae. New Series
publication date: 1992-12-15T00:00:00Z
review text: Definition. (a) A set \(X<\mathbb{R}^ k\) is 1-rectifiable if it is the image of a bounded subset \(U\subset\mathbb{R}\) under a Lipschitz continuous mapping \(f:U\to\mathbb{R}^ k\). (b) \(X\) is \(({\mathcal H}^ 1,1)\)-rectifiable if \({\mathcal H}^ 1(X)<\infty\) and \({\mathcal H}^ 1\)-almost all of \(X\) can be covered by a countable union of 1-rectifiable sets. \({\mathcal H}^ 1\) denotes here a 1-dimensional Hausdorff measure. The main result of the paper is Theorem. If \(X\) is a compact \(({\mathcal H}^ 1,1)\)-rectifiable subset of the unit sphere \(S\subset\mathbb{C}^ n\) such that the origin \(0\in\mathbb{C}^ n\) belongs to the polynomial hull \(\widehat X\), then \({\mathcal H}^ 1(X)\geq 2\pi\).
Mathematics Subject Classification ID: 32E20
zbMATH DE Number: 78086
zbMATH Keywords: set in the sphere
zbMATH Keywords: polynomial hull
reviewed by: S. Ivashkovich
MaRDI profile type: Publication
full work available at URL: 0019-3577(92)90004-5
OpenAlex ID: W2134827137
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description: scientific article; zbMATH DE number 78086