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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Lipschitz images with fractal boundaries
and their small surface wrapping

Author: Zoltán Buczolich
Journal: Proc. Amer. Math. Soc. 126 (1998), 3589-3595
MSC (1991): Primary 28A75; Secondary 28A80, 26B35
MathSciNet review: 1459112
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Abstract: Assume $E\subset H\subset \mathbf{R}^{m}$ and $\Phi :E\to \mathbf{R}^{m}$ is a Lipschitz $L$-mapping; $|H|$ and $||H||$ denote the volume and the surface area of $H$. We verify that there exists a figure $F\supset \Phi (E)$ with $||F||\leq c_{L} ||H||$, and, of course, $|F|\leq c_{L} |H|$, where $c_{L}$ depends only on the dimension and on $L$. We also give an example when $E=H\subset \mathbf{R}^{2}$ is a square and $||\Phi (E)||=\infty $; in fact, the boundary of $\Phi (E)$ can contain a fractal of Hausdorff dimension exceeding one.

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Additional Information

Zoltán Buczolich
Affiliation: Eötvös Loránd University, Department of Analysis, Budapest, Múzeum krt 6-8, H-1088, Hungary

Keywords: Lipschitz mapping, surface, fractal
Received by editor(s): January 31, 1997
Received by editor(s) in revised form: April 21, 1997
Additional Notes: This research was supported by the Hungarian National Foundation for Scientific Research, Grant Nos. T 019476 and T 016094
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society