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Continuously parametrized Besicovitch sets in R^n
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Järvenpää, Esa, Järvenpää, Maarit, Keleti, Tamás and Máthé, András (2011) Continuously parametrized Besicovitch sets in R^n. Annales Academiae Scientiarum Fennicae Mathematica , Vol.36 (No.2). pp. 411-421. doi:10.5186/aasfm.2011.3639 ISSN 1239629X.
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Official URL: http://dx.doi.org/10.5186/aasfm.2011.3639
Abstract
We study continuous 1-dimensional time parametrization and (n - 1)-dimensional direction parametrization of Besicovitch sets in R(n). In the 1-dimensional case we prove that for n >= 3 one can move a unit line segment (in fact even a full line) continuously in R(n) within a set of measure zero in such a manner that the line segment points in all possible directions. We also show that in R(n), for any n >= 2, one can parametrize unit line segments continuously by their direction so that all segments are contained in a set of arbitrarily small measure. However, if we parametrize lines continuously by their direction then the set which is not covered by their union is bounded.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Annales Academiae Scientiarum Fennicae Mathematica | ||||
Publisher: | Suomalainen Tiedeakatemia | ||||
ISSN: | 1239629X | ||||
Official Date: | 2011 | ||||
Dates: |
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Volume: | Vol.36 | ||||
Number: | No.2 | ||||
Number of Pages: | 11 | ||||
Page Range: | pp. 411-421 | ||||
DOI: | 10.5186/aasfm.2011.3639 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Funder: | Centre of Excellence in Analysis and Dynamics Research, Academy of Finland , Hungarian Scientific Foundation | ||||
Grant number: | 72655 (Hungarian Scientific Foundation ) |
Data sourced from Thomson Reuters' Web of Knowledge
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