Minkowski Content and local Minkowski Content for a class of self-conformal sets

Abstract

We investigate (local) Minkowski measurability of ${\mathcal{C}}^{1+\alpha }$ images of self-similar sets. We show that (local) Minkowski measurability of a self-similar set $K$ implies (local) Minkowski measurability of its image $F$ and provide an explicit formula for the (local) Minkowski content of $F$ in this case. A counterexample is presented which shows that the converse is not necessarily true. That is, $F$ can be Minkowski measurable although $K$ is not. However, we obtain that an average version of the (local) Minkowski content of both $K$ and $F$ always exists and also provide an explicit formula for the relation between the (local) average Minkowski contents of $K$ and $F$.

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References in corpus (1)

• Fractal curvature measures and Minkowski content for one-dimensional self-conformal sets

Cited by in corpus (4)

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• Localization results for Minkowski contents
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