Annales Academić Scientiarum Fennicć
Mathematica
Volumen 36, 2011, 411-421
University of Oulu, Department of Mathematical Sciences
P.O. Box 3000, 90014 University of Oulu, Finland; Esa.Jarvenpaa 'at' oulu.fi
University of Oulu, Department of Mathematical Sciences P.O. Box 3000, 90014 University of Oulu, Finland; Maarit.Jarvenpaa 'at' oulu.fi
Eötvös Loránd University, Department of Analysis
Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary; elek 'at' cs.elte.hu
University of Warwick, Department of Mathematics
Coventry CV4 7AL, United Kingdom; A.Mathe 'at' warwick.ac.uk
Abstract. We study continuous 1-dimensional time parametrization and (n - 1)-dimensional direction parametrization of Besicovitch sets in Rn. In the 1-dimensional case we prove that for n \ge 3 one can move a unit line segment (in fact even a full line) continuously in Rn within a set of measure zero in such a manner that the line segment points in all possible directions. We also show that in Rn, for any n \ge 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.
2010 Mathematics Subject Classification: Primary 28A75, 51M15, 51M25.
Key words: Besicovitch set, Kakeya needle problem, continuous parametrization, measure, lines in every direction.
Reference to this article: E. Järvenpää, M. Järvenpää, T. Keleti and A. Máthé: Continuously parametrized Besicovitch sets in Rn. Ann. Acad. Sci. Fenn. Math. 36 (2011), 411-421.
doi:10.5186/aasfm.2011.3639
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