Lipschitz images with fractal boundaries and their small surface wrapping
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- by Zoltán Buczolich PDF
- Proc. Amer. Math. Soc. 126 (1998), 3589-3595 Request permission
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.References
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Washek F. Pfeffer, The Gauss-Green theorem, Adv. Math. 87 (1991), no. 1, 93–147. MR 1102966, DOI 10.1016/0001-8708(91)90063-D
- D. E. G. Hare, An extension of a structure theorem of Bourgain, J. Math. Anal. Appl. 147 (1990), no. 2, 599–603. MR 1050230, DOI 10.1016/0022-247X(90)90373-N
Additional Information
- Zoltán Buczolich
- Affiliation: Eötvös Loránd University, Department of Analysis, Budapest, Múzeum krt 6-8, H-1088, Hungary
- Email: buczo@cs.elte.hu
- 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
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3589-3595
- MSC (1991): Primary 28A75; Secondary 28A80, 26B35
- DOI: https://doi.org/10.1090/S0002-9939-98-04433-5
- MathSciNet review: 1459112