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Asymptotic shape of small cells
Author(s) -
Beermann Mareen,
Redenbach Claudia,
Thäle Christoph
Publication year - 2014
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200328
Subject(s) - perimeter , mathematics , tessellation (computer graphics) , infinity , degenerate energy levels , poisson distribution , contrast (vision) , mathematical analysis , geometry , zero (linguistics) , scale (ratio) , distribution (mathematics) , statistics , physics , optics , linguistics , philosophy , quantum mechanics
A stationary Poisson line tessellation is considered whose directional distribution is concentrated on two different atoms with some positive weights. The shape of the typical cell of such a tessellation is studied when its area or its perimeter tends to zero. In contrast to known results where the area or the perimeter tends to infinity, it is shown that the asymptotic shape of cells having small area is degenerate. Again in contrast to the case of large cells, the asymptotic shape of cells with small perimeter is not uniquely determined. The results are accompanied by a large scale simulation study.