Mean Lipschitz-Killing curvatures for homogeneous random fractals

TL;DR

This paper investigates the asymptotic behavior of mean Lipschitz-Killing curvatures of homogeneous random fractals' parallel sets as the parallel radius approaches zero, extending deterministic results to a probabilistic setting.

Contribution

It introduces a framework for analyzing mean Lipschitz-Killing curvatures of homogeneous random fractals and derives their limit behaviors under specific geometric conditions.

Findings

Rescaled limits of mean Lipschitz-Killing curvatures exist as the radius tends to zero.

Integral representations for these limits are established, extending deterministic cases.

The results apply under the Uniform Strong Open Set Condition and additional geometric assumptions.

Abstract

Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the Uniform Strong Open Set Condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to zero. Moreover, integral representations are derived for these limits which extend those known in the deterministic case.