Various methods may be used to define the Minkowski-Bouligand dimension of a compact subset E in the plane. The best known is the box method. After introducing the notion of ε-connected set , we consider a new method based upon coverings of with crosses of diameter 2ε. To prove that this cross method gives the fractal dimension for all E, the main argument consists in constructing a special pavement of the complementary set with squares. This method gives rise to a dimension formula using integrals, which generalizes the well known variation method for graphs of continuous functions.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-2-5,
author = {Claude Tricot},
title = {A method for evaluating the fractal dimension in the plane, using coverings with crosses},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {181-199},
zbl = {1005.28006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-2-5}
}
Claude Tricot. A method for evaluating the fractal dimension in the plane, using coverings with crosses. Fundamenta Mathematicae, Tome 173 (2002) pp. 181-199. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-2-5/