## L. Ambrosio - V. Capasso - E. Villa

# On the approximation of geometric densities of random closed sets

created by ambrosio on 09 May 2006

[BibTeX]

*Submitted Paper*

**Inserted:** 9 may 2006

**Year:** 2006

**Abstract:**

Many real phenomena may be modelled as random closed sets in
$R^d$ of different Hausdorff dimensions. The authors
have recently revisited the concept of mean geometric
densities of random closed sets $\Theta_n$ with Hausdorff
dimension $n\leq d$ with respect to the standard Lebesgue measure
on $R^d$, in terms of expected values of a suitable class
of linear functionals (Delta functions \`{a} la Dirac). In
many real applications such
as fiber processes, $n$-facets of random tessellations
of dimension $n \leq d$ in spaces of dimension $d \geq 1,$
several problems are related to the
estimation of such mean densities; in order to face such
problems in the general setting of spatially inhomogeneous
processes, we suggest and analyze here an approximation of mean
densities for sufficiently regular random closed sets. We will
show how some known results in literature follow as particular
cases. A series of examples throughout the paper are provided to
exemplify various relevant situations.

**Keywords:**
Stochastic geometry, Random Distributions, Mean Geometric densities

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