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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Koch snowflake curve is tube-null


Author: Viktor Harangi
Journal: Proc. Amer. Math. Soc. 139 (2011), 1375-1381
MSC (2010): Primary 28A12, 28A80
Published electronically: December 1, 2010
MathSciNet review: 2748429
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Abstract: We show that the Koch curve is tube-null; that is, it can be covered by strips of arbitrarily small total width. In fact, we prove the following stronger result: the Koch curve can be decomposed into three sets such that each can be projected to a line in such a way that the image has Hausdorff dimension less than $ 1$. The proof contains geometric, combinatorial, algebraic and probabilistic arguments.


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Additional Information

Viktor Harangi
Affiliation: Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary
Email: bizkit@cs.elte.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10712-8
PII: S 0002-9939(2010)10712-8
Received by editor(s): April 15, 2010
Published electronically: December 1, 2010
Additional Notes: The author was supported by OTKA grant 72655.
Communicated by: Tatiana Toro
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.