Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 565-581

LARGE POROSITY AND DIMENSION OF SETS IN METRIC SPACES

Tapio Rajala

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; tamaraja 'at' jyu.fi

Abstract. We prove an asymptotically sharp dimension estimate for sets with large porosity in a collection of metric spaces. This generalizes a dimension estimate first proven by Salli. From the metric space we assume, among other properties, that it can be locally mapped into Rⁿ in a way that allows us to use Euclidean projections. We show that Rⁿ with any norm satisfies these conditions as well as every step two Carnot group. We also discuss the necessity of the conditions by examining various metric spaces where the estimates fail.

2000 Mathematics Subject Classification: Primary 28A80; Secondary 51F99, 54E35.

Key words: Normed vector spaces, Heisenberg group, porosity, packing dimension, Minkowski dimension.

Reference to this article: T. Rajala: Large porosity and dimension of sets in metric spaces. Ann. Acad. Sci. Fenn. Math. 34 (2009), 565-581.

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