Mean Minkowski and $s$-contents of $V$-variable random fractals

For general $V$-variable random fractals satisfying the Uniform Strong Open Set condition we show that the (average) mean $D$-dimensional Minkowski content exists, where the corresponding Minkowski dimension $D$ in the mean sense is, in general, greater than its almost sure variant. Moreover, we show that the (average) mean Minkowski content agrees with the (average) mean $D$-dimensional surface area based content and derive some integral representations. The latter are structurally the same as in different former cases of random fractals, and $D$ agrees with the dimensions obtained there.

Keywords

random fractals, Minkowski content, $s$-content, mean values

2010 Mathematics Subject Classification

28A75, 28A80, 60D99, 60G57, 60J85

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To the memory of Ka-Sing Lau

This work was supported by DFG Grant ZA 242/8-1.

Received 21 February 2022

Accepted 16 August 2022

Published 7 August 2024