The convergence of hulls of curves View Full Text


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Article Info

DATE

2022-03-18

AUTHORS

Alexander J. Izzo, Edgar Lee Stout

ABSTRACT

It is shown that a simple closed curve in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^n$$\end{document} that is a uniform limit of rectifiable simple closed curves each of which has nontrivial polynomial hull has itself nontrivial polynomial hull. In case the limit curve is rectifiable, the hull of the limit is shown to be the limit of the hulls. It is also shown that every rectifiable simple closed curve in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^n$$\end{document}, n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}, can be approximated in total variation norm by a polynomially convex, rectifiable simple closed curve that coincides with the original curve except on an arbitrarily small segment. As a corollary, it is shown that every rectifiable arc in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^n$$\end{document}, n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}, is contained in a polynomially convex, rectifiable simple closed curve. More... »

PAGES

1-16

References to SciGraph publications

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  • 1994-09. Approximation by automorphisms on smooth submanifolds of Cn in MATHEMATISCHE ANNALEN
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    http://scigraph.springernature.com/pub.10.1007/s00209-022-02972-2

    DOI

    http://dx.doi.org/10.1007/s00209-022-02972-2

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