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Isoperimetric problems and roses of neighborhood for stationary flat processes
Author(s) -
Spodarev Eugene
Publication year - 2003
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.200310032
Subject(s) - mathematics , intersection (aeronautics) , isoperimetric inequality , estimator , combinatorics , mathematical analysis , statistics , engineering , aerospace engineering
The paper yields necessary conditions for the directional distributions of stationary k –flat processes in ℝ d that maximize their intersection density of order 2, that is, the mean (2 k – d )–volume of their self–intersections in an observation window of unit d –volume. The conditions are given in terms of the rose of intersections (i.e., the intensity of intersections of the flat process with test flats). The notion of the rose of neighborhood is introduced which is an analogue of the rose of intersections for lower dimensional flat processes. Some properties of the rose of neighborhood are studied and an asymptotically unbiased estimator is given.