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. ISSN 1239629X

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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
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Annales Academiae Scientiarum Fennicae Mathematica
Publisher:	Suomalainen Tiedeakatemia
ISSN:	1239629X
Date:	2011
Volume:	Vol.36
Number:	No.2
Number of Pages:	11
Page Range:	pp. 411-421
Identification Number:	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 )
URI:	http://wrap.warwick.ac.uk/id/eprint/38943

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