Continuously parametrized Besicovitch sets in R^n
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 1239629XFull text not available from this repository.
Official URL: http://dx.doi.org/10.5186/aasfm.2011.3639
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|
|Number of Pages:||11|
|Page Range:||pp. 411-421|
|Funder:||Centre of Excellence in Analysis and Dynamics Research, Academy of Finland , Hungarian Scientific Foundation|
|Grant number:||72655 (Hungarian Scientific Foundation )|
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