# Mean Lipschitz-Killing curvatures for homogeneous random fractals

Jan Rataj Jan Rataj, Charles University, Faculty of Mathematics and Physics,
Sokolovská 83, 186 75 Praha 8, Czech Republic ,  Steffen Winter Steffen
Winter, Institute of Stochastics, Karlsruhe Institute of Technology,
Englerstr. 2, D-76131 Karlsruhe, Germany and  Martina Zähle Martina Zähle,
Institute of Mathematics, University of Jena, Ernst-Abbe-Platz 2, D-07743
Jena, Germany.

###### 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 $0$.
Moreover, integral representations are derived for these limits which extend
those known in the deterministic case.

###### Key words and phrases:

homogeneous random fractals, fractal curvatures, mean values

###### 2010 Mathematics Subject Classification:

Primary: 28A80, 60G57; Secondary: 28A75, 53C65, 60D05

Steffen Winter was supported by DFG grant WI 3264/5-1 and Martina Zähle by DFG
grant ZA 242/8-1.

##  1\. Introduction

Fractal versions of the $k$-th order Lipschitz-Killing curvatures $C_{k}$ in
$\mathbb{R}^{d}$ known from convex geometry, differential geometry and
geometric measure theory have been considered in [Wi08, WZ13] and subsequent
papers for deterministic self-similar sets and in [Za11] for random recursive
constructions. Compared to the latter class of models, homogeneous random
fractals, as first considered in [Ha92] (for a special case), possess more
dependencies in their structure. This leads, in particular, to the phenomenon
that their a.s. Minkowski dimension, which agrees again with the a.s.
Hausdorff dimension, is determined by a different equation than in the
recursive case (see [RU11] together with [Tr17] for the general case).
Moreover, in [Tr21a] it is shown that in the non-deterministic case there
exists no gauge function for an exact Hausdorff measure. It was conjectured in
[Za20] that the almost sure Minkowski content does not exist either. Meanwhile
this has been proved in [Tr21b]. In contrast to this striking difference in
the almost sure behaviour of both models, it turned out that for expectations
the results are the same as in the case of stochastically self-similar sets,
i.e., random recursive constructions (see [Za20]).

The Minkowski content may be viewed as the marginal case $k=d$ of the fractal
Lipschitz-Killing curvatures. It is the aim of the present paper to extend the
above mentioned results regarding Minkowski contents of homogeneous random
fractals $F$ to all Lipschitz-Killing curvatures.

For a deterministic self-similar set $F$ satisfying the open set condition and
some additional assumptions, Lipschitz-Killing curvatures can be introduced by
approximation with parallel sets. If $r>0$ is a regular value of the distance
function $d(\cdot,F)$ of $F$, then the $r$-parallel set
$F(r):=\\{x:\,d(x,F)\leq r\\}$ has ‘nice’ geometric properties. (More
precisely, its boundary is a Lipschitz manifold and the closure of its
complement has positive reach.) In particular, $F(r)$ admits Lipschitz-Killing
curvature measures $C_{k}(F(r),\cdot)$, $k=0,\dots,d-1$. We use the notation
$C_{k}^{\operatorname{var}}(F(r),\cdot)$ for the variation measures, and
$C_{k}(F(r)):=C_{k}(F(r),\mathbb{R}^{d})$ for the total mass. Provided that
almost all $r>0$ are regular values (which is always the case if $d\leq 3$,
see [Fu85]), the (total) _fractal Lipschitz-Killing curvatures_ of $F$ can be
introduced as rescaled (essential or averaged) limits

$\operatorname{ess\,lim}_{\varepsilon\rightarrow
0}\limits\varepsilon^{D-k}\,C_{k}(F(\varepsilon))\qquad\text{ or
}\qquad\lim_{\delta\rightarrow
0}\frac{1}{|\ln\delta|}\int_{\delta}^{1}\varepsilon^{D-k}\,C_{k}(F(\varepsilon))~{}\varepsilon^{-1}d\varepsilon,$

where $D$ is the Minkowski dimension of $F$. It turned out that the first one
(essential limit) exists in the case of non-lattice self-similar sets, while
the second one (averaged limit) exists in general, see [Wi08] and [Za11] for
details.

In order to give some idea of the results obtained in the paper, we give an
informal description of the considered model, a detailed definition can be
found in Section 3. Consider a random iterated function system (random IFS)
$f=(f_{1},\dots,f_{N})$ consisting of a random number $N\geq 2$ of contracting
similarities $f_{1},\dots,f_{N}$ with (random) contraction ratios
$r_{1},\dots,r_{N}$ fulfilling $0<r_{\min}\leq r_{i}\leq r_{\max}<1$ for some
deterministic values $r_{\min},r_{\max}$. We assume the _Uniform Open Set
Condition_ (UOSC): there exists a (deterministic) nonempty open set $O$ such
that almost surely, $f_{i}(O)\subset O$ and $f_{i}(O)\cap f_{j}(O)=\emptyset$,
$i\neq j\leq N$ (cf. (3.3)). The _homogeneous random fractal_ $F$ is defined
by means of an i.i.d. random sequence $(f^{n})_{n\in\mathbb{N}}$, where each
random IFS $f^{n}=(f^{n}_{1},\dots,f^{n}_{N^{n}})$ has the same distribution
as $f$. Then $F$ is obtained by applying at each construction step
$n\in\mathbb{N}$ the same chosen IFS $f^{n}$ to all components of that step.
More precisely, denoting
$\Sigma_{n}:=\\{\sigma_{1}\sigma_{2}\dots\sigma_{n}:\,1\leq\sigma_{i}\leq
N^{i},1\leq i\leq n\\}$, $n\in\mathbb{N}$, and
$f_{\sigma}:=f^{1}_{\sigma_{1}}\circ\dots\circ f^{n}_{\sigma_{n}}$,
$\sigma\in\Sigma_{n}$, the homogeneous random fractal associated to the random
IFS $f$ is the random set $F$ defined by

$F:=\bigcap_{n=1}^{\infty}\bigcup_{\sigma\in\Sigma_{n}}f_{\sigma}(\overline{O}).$

If $\mathbb{E}N<\infty$ (which we assume here throughout), then the equation
$\mathbb{E}\sum_{i=1}^{N}r_{i}^{D}=1$ has a unique solution $D\in[0,d]$,
called the _mean Minkowski dimension_ of $F$. (Note that it is given by the
same formula as in the random recursive case, cf. [Za20] and (3.9) below.) We
consider the measure $\mu:=\mathbb{E}\sum_{i=1}^{N}\mathbf{1}\\{|\ln
r_{i}|\in\cdot\\}r_{i}^{D}$ and its mean value
$\eta:=\mathbb{E}\sum_{i=1}^{N}|\ln r_{i}|\,r_{i}^{D}.$ Further, we let
$\Sigma_{*}:=\bigcup_{n=1}^{\infty}\Sigma_{n}$, and for any $r>0$, we denote
by $\Sigma(r)$ the set of all finite words
$\sigma=\sigma_{1}\dots\sigma_{n}\in\Sigma_{*}$ such that
$r^{1}_{\sigma_{1}}\cdots
r^{n}_{\sigma_{n}}\leq\frac{r}{2|O|}<r^{1}_{\sigma_{1}}\cdots
r^{n-1}_{\sigma_{n-1}}$. Here $|O|$ denotes the diameter of $O$. For
$\sigma\in\Sigma_{*}$ let $O_{\sigma}:=f_{\sigma}(O)$. Observe that
$O_{\sigma}$ is open, while its parallel sets $O_{\sigma}(r)$, $r\geq 0$, are
closed.

Our main result Theorem 3.2 establishes the existence of fractal Lipschitz-
Killing curvatures for a large class of homogeneous random fractals. The
following statement is a special case, as can easily be seen from Proposition
5.2 (d). It provides this existence under a stronger but easier to state
hypothesis.

######  Theorem 1.1.

Let $F\subset\mathbb{R}^{d}$ be a homogeneous random fractal with
$\mathbb{E}N<\infty$ satisfying UOSC with some open set
$O\subset\mathbb{R}^{d}$ such that $F\cap O\neq\emptyset$ almost surely. Let
$k\in\\{0,1,\ldots,d\\}$ and assume further that

  1. (i)

with probability one, almost all $r>0$ are regular values of $d(\cdot,F)$;

  2. (ii)

there exists a constant $c>0$ such that with probability one,

$C_{k}^{\operatorname{var}}\left(F(r),O_{\sigma}(r)\cap
O_{\sigma^{\prime}}(r)\right)\leq cr^{k},\text{ for a.a. }r>0\text{ and all
}\sigma,\sigma^{\prime}\in\Sigma(r)\text{ with }\sigma\neq\sigma^{\prime}.$

Define
$F_{i}:=\bigcap_{n=1}^{\infty}\bigcup_{\sigma\in\Sigma_{n},\sigma_{1}=i}f_{\sigma}(\overline{O})$
for $i\in\\{1,\ldots,N\\}$ and

$R_{k,1}(r):=\mathbb{E}C_{k}(F(r))-\mathbb{E}\sum_{i=1}^{N}\mathbf{1}_{(0,r_{i}]}(r)\,C_{k}\big{(}F_{i}(r)\big{)},\quad
r>0.$

Then,

$\overline{C}_{k,F}^{\operatorname{frac}}:=\lim_{\delta\rightarrow
0}\frac{1}{|\ln\delta|}\int_{\delta}^{1}\varepsilon^{D-k}\mathbb{E}\,C_{k}(F(\varepsilon))~{}\varepsilon^{-1}d\varepsilon=\frac{1}{\eta}\int_{0}^{1}r^{D-k-1}R_{k,1}(r)\,dr.$

Moreover, if the measure $\mu$ is non-lattice, then

$C_{k,F}^{\operatorname{frac}}:=\operatorname{ess\,lim}_{\varepsilon\rightarrow
0}\limits\varepsilon^{D-k}\mathbb{E}\,C_{k}(F(\varepsilon))=\overline{C}_{k,F}^{\operatorname{frac}}.$

As in the deterministic case, the values
$\overline{C}_{k,F}^{\operatorname{frac}}$ (and
$C_{k,F}^{\operatorname{frac}}$, in case they exist) can be interpreted as the
_mean (averaged) fractal Lipschitz-Killing curvatures_ of $F$ of order $k$.
(Note that in contrast to common differential geometric terminology in our
paper the term ‘mean’ curvatures is used in the probabilistic sense of
expectations.) For the cases $k=d-1$ and $k=d$, conditions (i) and (ii) are in
fact not needed.

Our result covers the case of deterministic self-similar sets considered in
[Wi08] and [Za11]. As a nontrivial example, we discuss below a family of
random homogeneous fractals, for which the associated random IFS choose
between two deterministic IFS generating Sierpiński type gaskets, see Section
5. In this case, the values of $C_{k,F}^{\operatorname{frac}}$ are determined
explicitly.

The plan of the paper is as follows. In Section 2 we give a brief survey on
the notion of curvature measures of parallel sets with references to the
literature. The measurability and continuity properties in Lemma 2.3 and Lemma
2.4 are used in the sequel for the probabilistic approach and application of
the classical Renewal theorem. But they are also of independent interest, e.g.
for possible extensions to other classes of random fractals as considered in
[RU11], [BHS12], and [Tr17].

Section 3 contains the construction of homogeneous random fractals and the
statement of the main result (Theorem 3.2 together with Remarks 3.3, 3.4 and
3.5). The proof of Theorem 3.2 in Section 4 is split into several steps. As in
the case of self-similar fractals and random recursive constructions the
Renewal theorem is a main tool. Most effort is needed for verifying the
corresponding conditions. It turns out that the integral representations for
the above limits derived here for homogeneous random fractals are very similar
to those known for (random) self-similar sets.

In Section 5, the hypothesis of Theorem 3.2 is replaced in part by a simpler
but stronger condition, cf. Proposition 5.2 and Theorem 1.1, which is easier
to verify in concrete examples. This is then demonstrated for a family of
homogeneous random Sierpiński gaskets as mentioned above.

In the Appendix the continuity of the curvature measures of parallel sets at
regular distances is proved, cf. Theorem 6.1, which is of independent
interest. It completes, in particular, the proof of Lemma 2.3.4 in [Za11] and
can be applied to other deterministic and stochastic models.

##  2\. Curvature measures of parallel sets

###  2.1. Definitions and relevant properties

For certain classes of compact sets $K\subset\mathbb{R}^{d}$ (including many
classical singular sets) it turns out that for Lebesgue-almost all distances
$r>0$ the parallel set

(2.1) $K(r):=\\{x\in\mathbb{R}^{d}:d(x,K)=\min_{y\in K}|x-y|\leq r\\}$

possesses the property that the closure of its complement

(2.2) $\widetilde{K(r)}:=\overline{K(r)^{c}}$

is a set with positive reach with Lipschitz boundary. (Recall that
$X\subset\mathbb{R}^{d}$ is a set with positive reach, if for some $\delta>0$
every point $x\in X(\delta)$ has a unique point $\Pi_{X}(x)\in X$ nearest to
$x$.) A sufficient condition is that $r$ is a regular value of the Euclidean
distance function to $K$ (see Fu [Fu85, Theorem 4.1] together with [RZ03,
Proposition 3]). In this case both sets $\widetilde{K(r)}$ and $K(r)$ are
Lipschitz $d$-manifolds of bounded curvature in the sense of [RZ05], implying
that their k-th Lipschitz-Killing curvature measures, $k=0,1,\ldots,d-1$, are
determined in this general context and agree with the classical versions in
the special cases. (See also Chapter 9 in [RZ19] for some background and
extensions.) Moreover, they satisfy

(2.3)
$C_{k}(K(r),\cdot)=(-1)^{d-1-k}C_{k}\big{(}\widetilde{K(r)},\cdot\big{)}\,,$

which implies that the $C_{k}(K(r),\cdot)$ are signed measures with finite
variation measures $C_{k}^{\operatorname{var}}(K(r),\cdot)$ and, moreover,
that they have explicit integral representations which reduce to the ones in
[Za86b] (cf. [RZ05, Theorem 3] for the general case). The corresponding normal
cycle representation will briefly be mentioned below. We list some of the main
properties of curvature measures of parallel sets which will be used
repeatedly: $2C_{d-1}(K(r),\cdot)$ agrees with $(d-1)$-dimensional Hausdorff
measure $\mathcal{H}^{d-1}$ on the boundary $\partial K(r)$. The latter is a
finite measure for all $r>0$ and all compact sets $K$. Therefore we use the
notation

$C_{d-1}(K(r),\cdot):=\frac{1}{2}\mathcal{H}^{d-1}(\partial K(r)\cap(\cdot))$

in the general case. For completeness we also define $C_{d}(K(r),\cdot)$ to be
the Lebesgue measure restricted to $K(r)$. The total curvatures of $K(r)$ are
denoted by

(2.4) $C_{k}(K(r)):=C_{k}(K(r),\mathbb{R}^{d})\,,~{}~{}k=0,\ldots,d\,.$

By an associated Gauss-Bonnet theorem (see [RZ03, Theorems 2,3]) the Gauss
curvature $C_{0}(K(r))$ coincides with the Euler-Poincaré characteristic
$\chi(K(r))$, whenever the curvature measure $C_{0}(K(r),\cdot)$ is defined as
described above. For smooth boundaries $\partial K(r)$ in the differential
geometric setting, $C_{k}(K(r))$ can be interpreted as the total $k$-th order
mean curvature of $K(r)$, which is extrinsic if $d-1-k$ is odd. For $k=d-2$ it
is also known as extrinsic total mean curvature of $K(r)$, and for $k=d-3$ it
coincides with the intrinsic total scalar curvature of $\partial K(r)$, up to
certain constants.

The curvature measures are motion invariant, i.e.,

(2.5) $C_{k}(g(K(r)),g(\cdot))=C_{k}(K(r),\cdot)~{}~{}\mbox{for any Euclidean
motion}~{}g\,,$

they are homogeneous of degree $k$, i.e.,

(2.6) $C_{k}\big{(}(\lambda K)(\lambda
r),\lambda(\cdot)\big{)}=C_{k}\big{(}\lambda
K(r),\lambda(\cdot)\big{)}=\lambda^{k}\,C_{k}(K(r),\cdot)\,,~{}~{}\lambda>0\,,$

and _locally determined_ , i.e.,

(2.7) $C_{k}(K(r),(\cdot)\cap G)=C_{k}(K^{\prime}(r^{\prime}),(\cdot)\cap G)$

for any open set $G\subset\mathbb{R}^{d}$ such that $K(r)\cap
G=K^{\prime}(r^{\prime})\cap G$, where $K(r)$ and $K^{\prime}(r^{\prime})$ are
both parallel sets such that the closures of their complements have positive
reach.

###  2.2. Unit normal cycles, measurability and continuity

We now summarize some facts about sets with positive reach needed in the
sequel:
Recall that $\operatorname{reach}X$ of a set $X\subset\mathbb{R}^{d}$ is
defined as the supremum over all $r\geq 0$ such that for every point $x$ in
the $r$-parallel set of $X$ there is a unique point $\Pi_{X}x\in X$ nearest to
$x$. The mapping $\Pi_{X}$ (on its domain) is called the _metric projection_
onto $X$. For a set $X$ of positive reach the _unit normal bundle_ is defined
as

$\operatorname{nor}X:=\\{(x,n)\in\mathbb{R}^{d}\times S^{d-1}:x\in
X,\,n\in\operatorname{Nor}(X,x)\\}$

where $\operatorname{Nor}(X,x)$ is the dual cone to the (convex) tangent cone
of $X$ at $x$.

If additionally $\operatorname{nor}X\cap\rho(\operatorname{nor}X)=\emptyset$
for the normal reflection $\rho\>(x,n)\mapsto(x,-n)$, then $X$ is a
$d$-dimensional Lipschitz manifold with boundary (see [RZ03, Proposition 3]).

For general $X$ with $\operatorname{reach}X>0$ there is an associated
rectifiable current called the unit normal cycle of $X$ which is given by

$N_{X}(\varphi):=\int_{\operatorname{nor}X}\langle
a_{X}(x,n),\varphi(x,n)\rangle\mathcal{H}^{d-1}(d(x,n))$

for an appropriate unit simple $(d-1)$-vector field
$a_{X}=a_{1}\wedge\ldots\wedge a_{d-1}$ associated a.e. with the tangent
spaces of $\operatorname{nor}X$ and for integrable differential $(d-1)$-forms
$\varphi$. In these terms for $k\leq d-1$ the curvature measure may be
represented by

$C_{k}(X,B)=N_{X}\llcorner\mathbf{1}_{B\times\mathbb{R}^{d}}(\varphi_{k})=\int_{\operatorname{nor}X\cap(B\times\mathbb{R}^{d})}\langle
a_{X}(x,n),\varphi_{k}(n)\rangle\mathcal{H}^{d-1}(d(x,n))$

for any bounded Borel set $B\subset\mathbb{R}^{d}$, where the $k$-th
Lipschitz-Killing curvature form $\varphi_{k}$ does not depend on the points
$x$ and is defined by its action on a simple $(d-1)$-vector
$\eta=\eta_{1}\wedge\ldots\wedge\eta_{d-1}$ as follows: Let $\pi_{0}(y,z):=y$
and $\pi_{1}(y,z):=z$ be the coordinate projections in
$\mathbb{R}^{d}\times\mathbb{R}^{d}$, $e^{\prime}_{1},\ldots,e^{\prime}_{d}$
be the dual basis of the standard basis in $\mathbb{R}^{d}$ and
$\mathcal{O}_{k}$ the surface area of the $k$-dimensional unit sphere. Then we
have

$\langle\eta,\varphi_{k}(n)\rangle:=\mathcal{O}_{d-k}^{-1}\sum_{\varepsilon_{i}\in\\{0,1\\},\,\sum\varepsilon_{i}=d-1-k}\langle\pi_{\varepsilon_{1}}\eta_{1}\wedge\ldots\wedge\pi_{\varepsilon_{d-1}}\eta_{d-1}\wedge
n,e^{\prime}_{1}\wedge\ldots\wedge e^{\prime}_{d}\rangle\,.$

Below we will use the following nice behavior of parallel sets with
sufficiently large distances. The diameter of a compact set $K$ is denoted by
$|K|$.

######  Lemma 2.1.

[Za11, Theorem 4.1] For any $R>\sqrt{2}$ and $k=0,1,\ldots,d$ there exists a
constant $c_{k}(R)$ such that for any compact set $K\subset\mathbb{R}^{d}$ and
any $r\geq R|K|$,

$\operatorname{reach}(\widetilde{K(r)})\geq|K|\sqrt{R^{2}-1}\,,$

$\partial K(r)$ is a $(d-1)$-dimensional Lipschitz submanifold, and

$\sup_{r\geq
R|K|}\frac{C_{k}^{\operatorname{var}}(K(r),\mathbb{R}^{d})}{r^{k}}\leq
c_{k}(R)\,.$

(It is well-known that for compact convex sets $K$ the last two properties
hold for all $r>0$, the last one is the sharper version of an isodiametric
inequality with an optimal constant.)

In the general case we define the set of regular pairs of compact sets and
distances by

(2.8)
$\operatorname{Reg}:=\bigg{\\{}(r,K)\in[0,\infty)\times\mathcal{K}:\widetilde{K(r)}\in\mathcal{PR},\,\operatorname{nor}\widetilde{K(r)}\cap\rho(\operatorname{nor}\widetilde{K(r)})=\emptyset\bigg{\\}}\,,$

where $\mathcal{K}$ denotes the family of all nonempty compact subsets of
$\mathbb{R}^{d}$, $\mathcal{PR}$ stands for the space of subsets of
$\mathbb{R}^{d}$ with positive reach and $\rho$ is the normal reflection given
by $(x,n)\mapsto(x,-n)$.

######  Remark 2.2.

An equivalent representation of regular pairs at positive distances is

(2.9)
$\operatorname{Reg}\cap((0,\infty)\times\mathcal{K})=\left\\{(r,K)\in(0,\infty)\times\mathcal{K}:\,r\text{
is a regular value of }d_{K}\right\\},$

where $d_{K}:x\mapsto d(x,K)$ is the distance function of $K$, see Appendix.

In Fu [Fu85, Theorem 4.1] it is shown that in space dimensions $d\leq 3$ for
any compact set $K$ there exists a bounded exceptional set $E$ of Lebesgue
measure $0$ such that for any $r\notin E$ the set $\widetilde{K(r)}$ has
positive reach and
$\operatorname{nor}\widetilde{K(r)}\cap\rho(\operatorname{nor}\widetilde{K(r)})=\emptyset$.
Moreover, if $r>\sqrt{d/(2d+2)}|K|$, these two properties hold for any space
dimension $d$. It follows that in space dimensions $d\leq 3$ for any compact
$K$ the pair $(r,K)$ is regular for Lebesgue-a.a. $r$. In higher dimensions we
will formulate this as a regularity condition on the random fractal sets.

For probabilistic purposes we need the following measurability properties. Let
$\mathcal{F}$ denote the space of all closed subsets of $\mathbb{R}^{d}$
provided with the Vietoris topology (generated by the sets
$\\{A\in\mathfrak{F}:A\cap O\neq\emptyset\\}$ and $\\{A\in\mathfrak{F}:A\cap
C=\emptyset\\}$ for open $O$ and closed $C$) and the associated Borel
$\sigma$-algebra $\mathfrak{B}(\mathcal{F})$. On its subspace $\mathcal{K}$
(of _nonempty compact_ sets), this topology can be metrized by the _Hausdorff
distance_

$d_{H}(K,L):=\max\left\\{\max_{x\in K}d(x,L),\max_{y\in
L}d(y,K)\right\\},\quad K,L\in\mathcal{K},$

and $\mathfrak{B}(\mathcal{K})$ denotes the Borel $\sigma$-algebra on
$\mathcal{K}$.

######  Lemma 2.3.

The following assertions hold.

  1. (i)

$\mathcal{PR}\in\mathfrak{B}(\mathfrak{F})$.

  2. (ii)

The mapping $(r,K)\mapsto\widetilde{K(r)}$ from $[0,\infty)\times\mathcal{K}$
to $\mathfrak{F}$ is
$[\mathfrak{B}([0,\infty))\otimes\mathfrak{B}(\mathcal{K})\,,\,\mathfrak{B}(\mathfrak{F})]$-measurable.

  3. (iii)

For any bounded Borel set $B\subset\mathbb{R}^{d}$ and $k=0,\dots,d$, the
mappings $X\mapsto C_{k}(X,B)$ and $X\mapsto C_{k}^{\operatorname{var}}(X,B)$
from $\mathcal{PR}$ to $\mathbb{R}$ are
$[\mathfrak{B}(\mathfrak{F})\cap\mathcal{PR},\mathfrak{B}(\mathbb{R})]$-mea-
surable.

  4. (iv)

For $k=0,\dots,d$, the mappings $(X,F)\mapsto C_{k}(X,F)$ and $(X,F)\mapsto
C^{\operatorname{var}}_{k}(X,F)$ from $\mathcal{PR}\times\mathcal{K}$ to
$\mathbb{R}$ are
$[(\mathfrak{B}(\mathfrak{F})\cap\mathcal{PR})\otimes\mathfrak{B}(\mathcal{K}),\mathfrak{B}(\mathbb{R})]$-measurable.

  5. (v)

$\operatorname{Reg}\in\mathfrak{B}([0,\infty))\otimes\mathfrak{B}(\mathcal{K})$.

###### Proof.

(i) See Proposition 1.1.1 in [Za86].

(ii) It is easy to see that for the centered balls $B(R)$ of radius $R$ the
mappings $(r,K)\mapsto\widetilde{K(r)}\cap B(R)$ are continuous (with respect
to the Hausdorff metric). Using that
$\widetilde{K(r)}=\bigcup_{R=1}^{\infty}(\widetilde{K(r)}\cap B(R))$ we infer
the assertion.

(iii) See Theorem 2.1.2 (i) and Theorem 6.2.2 in [Za86].

(iv) Due to (iii), the mappings $X\mapsto C_{k}(X,\cdot)$ and $X\mapsto
C_{k}^{\operatorname{var}}(X,\cdot)$ are random signed measures on the
probability space
$(\mathcal{PR},\mathfrak{B}(\mathfrak{F})\cap\mathcal{PR},\Pr)$ for any
probability measure $\Pr$ on the given space. The result then follows from
Lemma 6.5.

(v) This follows from (i), (ii), and the fact that the mapping
$X\mapsto\operatorname{nor}X$ from $\mathcal{PR}$ into the space of closed
subsets of $\mathbb{R}^{d}\times\mathbb{R}^{d}$ is measurable. ∎

In order to apply the classical Renewal theorem to the curvatures of random
fractals we additionally need the following continuity property. Recall that
the set of regular values of the distance function $d_{K}$ is open. (This can
e.g. be seen from (6.1) in the Appendix.) Therefore, there exists for each
regular value $r_{0}$ an $\varepsilon>0$ such that for any
$r\in(r_{0}-\varepsilon,r_{0}+\varepsilon)$, $r$ is a regular value of
$d_{K}$, too.

######  Lemma 2.4.

For any $(r_{0},K)\in\operatorname{Reg}$ with $r_{0}>0$ and any
$k\in\\{0,\ldots,d\\}$, the measures $C_{k}(K(r),\cdot)$ converge weakly to
$C_{k}(K(r_{0}),\cdot)$ as $r\to r_{0}$.

A detailed proof will be given in the Appendix, see Theorem 6.1.

######  Remark 2.5.

If $k=d$ or $k=d-1$, then the weak convergence in Lemma 2.4 remains valid for
general compact sets $K$ and almost all $r_{0}>0$, i.e., we need not restrict
$K$ to the class $\mathcal{PR}$ or $r_{0}$ to regular values of $K$. To see
this, note that $r\mapsto V_{d}(K(r))=\mathcal{L}^{d}(K(r))$ is a _Kneser
function_ (see [St76, Lemma 5]) and that
$V_{d-1}(K(r))=\mathcal{H}^{d-1}(\partial
K(r))=\frac{d}{dr}\mathcal{L}^{d}(K(r))$ for all $r>0$ up to a countable set
(see [RW10, Corollary 2.5]). The continuity of $V_{d}(r)$ at all $r>0$ and of
$V_{d-1}(K(r))$ at all $r>0$ up to a countable set follows then from the
properties of Kneser functions (see [St76, Lemma 2]).

Furthermore, since both mappings $K\mapsto\mathcal{L}^{d}(K)$ and
$K\mapsto\mathcal{H}^{d-1}(\partial K)$ are measurable on $\mathcal{K}$ (cf.
[Za82]), Lemma 2.3 (iii) and (iv) are valid for $k=d$ and $k=d-1$ with
$\mathcal{PR}$ replaced by the larger space $\mathcal{K}$.

##  3\. Construction of homogeneous random fractals and statement of the
results

For fixed $0<r_{\min}<r_{\max}<1$ let $\operatorname{Sim}$ be the set of all
contractive similarities $g:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ with
contraction ratios $r$ such that $r_{\min}\leq r\leq r_{\max}$. We equip
$\operatorname{Sim}$ with the topology given by uniform convergence on compact
sets. $\mathcal{B}$ denotes the associated Borel $\sigma$-algebra. The space
$\Omega_{0}:=\bigcup_{k=2}^{\infty}\operatorname{Sim}^{k}$ together with the
$\sigma$-algebra
$\mathcal{F}_{0}:=\big{\\{}A\subset\Omega_{0}:A\cap\operatorname{Sim}^{k}\in\otimes_{i=1}^{k}\mathcal{B}~{}\mbox{for
all}~{}k\geq 2\big{\\}}$, and with a distribution $\mathbb{P}_{0}$ on it
provide the primary probability space
$[\Omega_{0},\mathcal{F}_{0},\mathbb{P}_{0}]$. This space is used to generate
a random _iterated function system_ (IFS) $(f_{1},\ldots,f_{N})$, with a
random number $N$ of mappings $f_{1},\ldots,f_{N}$ chosen randomly from
$\operatorname{Sim}$. Note that, by construction, $N\geq 2$.

For the definition of the homogeneous model, we need a sequence of independent
and equally distributed random IFS. Therefore, the basic probability space for
the model is the product space

(3.1)
$[\Omega,\mathcal{F},\mathbb{P}]:=\bigotimes_{n=1}^{\infty}~{}[\Omega_{0},\mathcal{F}_{0},\mathbb{P}_{0}]$

and the expectation symbol $\mathbb{E}$ will be used for integration with
respect to $\mathbb{P}$.

The elements of $\Omega$ are denoted by

$\omega=(\omega_{1},\omega_{2},\ldots):=((f_{1}^{1},\ldots,f_{N^{1}}^{1}),(f_{1}^{2},\ldots,f_{N^{2}}^{2}),\ldots)\,,$

and $r_{i}^{n}$ are the contraction ratios of the similarities $f_{i}^{n}$.
For $f_{i}^{1}$, $r_{i}^{1}$ and $N^{1}$ we will often write $f_{i}$, $r_{i}$
and $N$, resp. Below we will use the measurable mapping

$\theta:\Omega\rightarrow\Omega,\quad\theta(\omega_{1},\omega_{2},\omega_{3},\ldots):=(\omega_{2},\omega_{3},\ldots)\,,$

and for a random element $X$ and $n\in\mathbb{N}$ we define the shifted random
element $X^{(n)}$ by

(3.2) $\displaystyle
X^{(n)}(\omega):=X(\theta^{n}\omega),\quad\omega\in\Omega\,,$

where $\theta^{n}$ is the $n$-fold application of the shift $\theta$. (If it
is clear from the context, the argument $\omega$ will be omitted.)

Note that, for each $n\in\mathbb{N}$, $(f_{1}^{n},\ldots,f_{N^{n}}^{n})$ is a
random IFS of random length $N^{n}$ with distribution $\mathbb{P}_{0}$
representing the $n$-th construction step. For different $n$ they are
independent of each other.

In the sequel we assume that the Uniform Open Set Condition (UOSC) is
satisfied: there exists a nonempty bounded open set $O\subset\mathbb{R}^{d}$
such that $\mathbb{P}_{0}$-a.s.

(3.3) $\bigcup_{i=1}^{N}f_{i}(O)\subset O\quad\mbox{and}\quad f_{i}(O)\cap
f_{j}(O)=\emptyset~{},i\neq j\,.$

Then with $\mathbb{P}$-probability 1 all IFS in the product space fulfill this
UOSC.

The corresponding random fractal set is introduced by means of a random coding
tree:
$\Sigma_{n}=\Sigma_{n}(\omega):=\\{\sigma_{1}\ldots\sigma_{n}:1\leq\sigma_{i}\leq
N^{i},\,i=1,\ldots,n\\}$ is the set of all nodes at level $n$ and
$\Sigma_{*}:=\bigcup_{n=0}^{\infty}\Sigma_{n}$ is the set of all nodes of the
tree, where $\Sigma_{0}$ denotes the empty code at level $0$.

Recall from (3.2) that $\Sigma_{l}^{(k)}$ is defined by
$\Sigma_{l}^{(k)}(\omega)=\Sigma_{l}(\theta^{k}\omega)$, $\omega\in\Omega$.
For $\sigma=\sigma_{1}\ldots\sigma_{k}\in\Sigma_{k}$ and
$\tau=\tau_{1}\ldots\tau_{l}\in\Sigma_{l}^{(k)}$ we write
$\sigma\tau:=\sigma_{1}\ldots\sigma_{k}\tau_{1}\ldots\tau_{l}\in\Sigma_{k+l}$
for the concatenation of these codes. If
$\sigma=\sigma_{1}\ldots\sigma_{n}\in\Sigma_{n}$ and $0\leq k\leq n$, then
$\sigma|k:=\sigma_{1}\ldots\sigma_{k}$ denotes the restriction to the first
$k$ entries of $\sigma$, and $|\sigma|:=n$ is the length of $\sigma$. For any
fixed $n=1,2,\ldots$, we associate to each $\sigma\in\Sigma_{n}$ the same
random IFS $\left(f_{1}^{n+1},\ldots,f_{N^{n+1}}^{n+1}\right)$. This leads to
the homogeneous structure. (In the $V$-variable case these random IFS are
chosen by means of $V$ different types. Here we have $V=1$, and in the case of
random recursive constructions, where $V=\infty$, for different
$\sigma\in\Sigma_{n}$ the IFS are i.i.d.) Furthermore, we define the random
mappings

$f_{\sigma}:=f_{\sigma_{1}}^{1}\circ f_{\sigma_{2}}^{2}\circ\dots\circ
f_{\sigma_{n}}^{n}$

with contraction ratios
$r_{\sigma}:=r_{\sigma_{1}}^{1}r_{\sigma_{2}}^{2}\ldots r_{\sigma_{n}}^{n}$.
Then the random compact set

(3.4)
$F=F(\omega):=\bigcap_{n=1}^{\infty}\bigcup_{\sigma\in\Sigma_{n}}f_{\sigma}(\overline{O})$

is $\mathbb{P}$-a.s. determined and measurable with respect to
$\mathcal{B}(\mathcal{K})$, the Borel $\sigma$-algebra determined by the
Hausdorff distance $d_{H}$ on the space $\mathcal{K}$ of nonempty compact
subsets of $\mathbb{R}^{d}$. It is called the associated homogeneous random
fractal. $F$ is stochastically self-similar in the following sense (recall
from (3.2) that $F^{(n)}(\omega)=F(\theta^{n}\omega)$):

$F=\bigcup_{i=1}^{N}f_{i}(F^{(1)})~{}~{},~{}~{}\mathbb{P}\,\text{- a.s.}~{}.$

More generally, for all $n\in\mathbb{N}$,

(3.5)
$F=\bigcup_{\sigma\in\Sigma_{n}}f_{\sigma}(F^{(n)})~{}~{},~{}~{}\mathbb{P}\,\text{-
a.s.}~{},$

where the random compact set $F^{(n)}$ is independent of the random mappings
$\\{f_{\sigma},\,\sigma\in\Sigma_{n}$} and has the same distribution as $F$.
We will also use the abbreviation

$F_{\sigma}:=f_{\sigma}(F^{(|\sigma|)})~{}~{}{\rm
for}~{}~{}~{}~{}\sigma\in\Sigma_{*}\,.$

For a boundedness condition in the application of the Renewal theorem we will
further use a formula similar to (3.5) with respect to some Markov stop: Fix
an arbitrary constant $R>\sqrt{2}|O|$ and define for all $0<r<R$ a random
subset of codes by

(3.6) $\Sigma(r):=\\{\sigma\in\Sigma_{*}:R\,r_{\sigma}\leq
r<R\,r_{\sigma||\sigma|-1}\\}\,,$

where, by convention, $r_{\sigma||\sigma|-1}=1$, if $|\sigma|=1$. It is
convenient to set $\Sigma(r):=\Sigma_{0}$ for $r\geq R$. Then we have

(3.7)
$F=\bigcup_{\sigma\in\Sigma(r)}F_{\sigma}~{}~{},~{}~{}\mathbb{P}\,\text{-
a.s.}~{}.$

In order to formulate the main results we also need the following random sets
of _boundary codes_ , i.e., codes $\sigma\in\Sigma(r)$ for which the parallel
set $F_{\sigma}(r)$ has distance less than $r$ to the boundary of the first
iterate $f(O):=\bigcup_{i=1}^{N}f_{i}(O)$ of the basic open set $O$ under the
random similarities:

(3.8)
$\Sigma_{b}(r):=\big{\\{}\sigma\in\Sigma(r):F_{\sigma}(r)\cap(f(O)^{c})(r)\neq\emptyset\big{\\}}\,,$

Our considerations below do not depend on the choice of the constant $R$ which
is related to Lemma 2.1, (where the above $R$ corresponds to $R|O|$).

In the sequel many relationships between random elements are fulfilled only
with probability $1$. We will not mention this, if it can be seen from the
context. Furthermore, the different meanings of $F=F(\omega)$ as random set
and $F(r)=F(\omega,r)$ as parallel set of the random set $F$ will also be
clear from the context.

The measurability properties of the random elements used in the sequel follow
from their definitions together with Lemma 2.3 and Remark 2.5.

Recall now that $2\leq N$ and assume that $\mathbb{E}N<\infty$. Let $D$ be the
number determined by

(3.9) $\mathbb{E}\sum_{i=1}^{N}r_{i}^{D}=1\,.$

Note that UOSC implies $D\leq d$.

(3.10) $\mu(\cdot):=\mathbb{E}\sum_{i=1}^{N}\mathbf{1}_{(\cdot)}(|\ln
r_{i}|)\,r_{i}^{D}$

is an associated probability distribution for the logarithmic contraction
ratios $r_{i}$ of the primary random IFS. The corresponding mean value is
denoted by

(3.11) $\eta:=\mathbb{E}\sum_{i=1}^{N}|\ln r_{i}|\,r_{i}^{D}.$

For our main result we need a slightly stronger condition than UOSC (3.3),
namely the Uniform Strong Open Set Condition (USOSC). It is satisfied, if

(3.12) $\mbox{UOSC holds for some $O$ such that}~{}~{}\mathbb{P}(F\cap
O\neq\emptyset)>0\,.$

######  Remark 3.1.

In the literature instead of $\mathbb{P}(F\cap O\neq\emptyset)>0$ the
condition

$\mathbb{P}(F\cap O\neq\emptyset)=1$

has been used. By the following arguments one can see that these conditions
are equivalent: If $\mathbb{P}(F\cap O\neq\emptyset)>0$, there must be some
$n_{0}\in\mathbb{N}$ such that the set

$S_{0}:=\\{\omega:\exists~{}\sigma\in\Sigma_{n_{0}}~{}~{}\mbox{with}~{}~{}f_{\sigma}(\overline{O})\subset
O\\}$

has positive probability. (Otherwise $F$ would concentrate on the boundary of
$O$.) Then the sets $S_{k}:=\theta^{kn_{0}}(S_{0})$, $k=0,1,2,\ldots$, are
independent and have all the same probability. Hence the Borel-Cantelli lemma
implies that $\mathbb{P}\big{(}\bigcap_{n=0}^{\infty}\bigcup_{k\geq
n}S_{k}\big{)}=1$, in particular, $\mathbb{P}(\bigcup_{k=0}^{\infty}S_{k})=1$.
This and UOSC lead to $\mathbb{P}(F\cap O\neq\emptyset)=1$.

By Lemma 2.1 the boundary $\partial F(r)$ of the parallel set
$F(r)=F(\omega,r)$ is a $(d-1)$-Lipschitz manifold and $\widetilde{F(r)}$ has
positive reach for all $r\geq R$, where $R$ is some constant such that
$R>\sqrt{2}|O|\geq\sqrt{2}|F|$. For $r<R$, we will use the following
regularity condition. (Recall from (2.8) the definition of the set
$\operatorname{Reg}$ of regular pairs.) The random set $F$ is called regular
if the measure of irregular pairs vanishes, i.e., if

(3.13)
$\mathbb{P}\times\mathcal{L}\big{(}\\{(\omega,r)\in\Omega\times(0,\infty):(r,F)\notin\operatorname{Reg}\\}\big{)}=0\,.$

In this case we also consider the set

$\operatorname{Reg}_{*}:=\\{(\omega,r):(r,F_{\sigma})\in\operatorname{Reg}~{}~{}\mbox{for
all}~{}\sigma\in\Sigma_{*}\\}\,$

(Recall that $\Sigma_{*}=\bigcup_{n=1}^{\infty}\Sigma_{n}$.) and note that

(3.14)
$\int\int\mathbf{1}_{(\operatorname{Reg}_{*})^{c}}(\omega,r)\mathcal{L}(dr)\mathbb{P}(d\omega)=0\,.$

(To see this, let
$\mathcal{N}(\omega):=\\{r>0:(r,F)\notin\operatorname{Reg}\\}$ and observe
that

$\displaystyle\int\int\mathbf{1}_{(\operatorname{Reg}_{*})^{c}}(\omega,r)\mathcal{L}(dr)\mathbb{P}(d\omega)=\int\mathcal{L}\big{(}\bigcup_{\sigma\in\Sigma_{*}}r_{\sigma}(\omega)\mathcal{N}(\theta^{|\sigma|}(\omega))\big{)}\mathbb{P}(d\omega)$
$\displaystyle\leq$
$\displaystyle\sum_{n=1}^{\infty}\int\sum_{\sigma\in\Sigma_{n}}\mathcal{L}\big{(}r_{\sigma}(\omega)\mathcal{N}(\theta^{n}(\omega)\big{)}\mathbb{P}(d\omega)=\sum_{n=1}^{\infty}\int\sum_{\sigma\in\Sigma_{n}}r_{\sigma}(\omega)\mathcal{L}(\mathcal{N}(\theta^{n}(\omega))\mathbb{P}(d\omega)$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\int\sum_{\sigma\in\Sigma_{n}}r_{\sigma}(\omega)\int\mathcal{L}(\mathcal{N}(\omega^{\prime}))\mathbb{P}(d\omega^{\prime})\mathbb{P}(d\omega)=0\,,$

since under the regularity condition (3.13) the inner integral vanishes. Here
we have used that the random sets $\mathcal{N}(\theta^{n}(\omega))$ are
independent of the events up to the step $n$ and have the same distribution as
$\mathcal{N}(\omega)$.)

Now we can formulate our main result. In the sequel, the occurring essential
limits and suprema are always meant with respect to Lebesgue-a.a. arguments.

######  Theorem 3.2.

Let $k\in\\{0,1,\ldots,d\\}$ and let $F$ be a homogeneous random fractal
satisfying the Uniform Strong Open Set Condition (3.12) with basic set
$O\subset\mathbb{R}^{d}$ and $\mathbb{E}N<\infty$. Let $R>\sqrt{2}|O|$. For
$k\leq d-2$ we additionally suppose the following:

  * (i)

if $d\geq 4$, then $F$ is regular in the sense of (3.13),

  * (ii)

for all $r_{0}\in(0,R)$,

$\mathbb{E}\operatorname{ess\,sup}_{r_{0}\leq r\leq
R}\limits\,\max_{\sigma\in\Sigma_{b}(r)}C_{k}^{\operatorname{var}}\bigg{(}F(r),\partial(F_{\sigma}(r))\cap\partial\big{(}\bigcup_{\sigma^{\prime}\in\Sigma(r),~{}\sigma^{\prime}\neq\sigma}F_{\sigma^{\prime}}(r)\big{)}\bigg{)}<\infty\,,$

  * (iii)

there is a constant $C>0$ such that with probability 1,

$\mathbb{E}\bigg{[}\max_{\sigma\in\Sigma_{b}(r)}\,r^{-k}\,C_{k}^{\operatorname{var}}\bigg{(}F(r),\partial(F_{\sigma}(r))\cap\partial\big{(}\bigcup_{\sigma^{\prime}\in\Sigma(r),~{}\sigma^{\prime}\neq\sigma}F_{\sigma^{\prime}}(r)\big{)}\bigg{)}\bigg{|}\sharp(\Sigma_{b}(r))\bigg{]}\leq
C$

for Lebesgue almost all $r\in(0,R]$.

Let $L>0$ and set for almost all $r>0$,

$R_{k,L}(r):=\mathbb{E}C_{k}(F(r))-\mathbb{E}\sum_{i=1}^{N}\mathbf{1}_{(0,Lr_{i}]}(r)\,C_{k}\big{(}F_{i}(r)\big{)}\,.$

Then the following assertions hold:

  * (I)

If the measure $\mu$ is non-lattice, then

$C_{k,F}^{\operatorname{frac}}:=\operatorname{ess\,lim}_{\varepsilon\rightarrow
0}\limits\varepsilon^{D-k}\mathbb{E}\,C_{k}(F(\varepsilon))=\frac{1}{\eta}\int_{0}^{L}r^{D-k-1}R_{k,L}(r)\,dr\,.$

  * (II)

If the measure $\mu$ is lattice with constant $c$, then for almost all
$s\in[0,c)$

$\lim_{n\rightarrow\infty}e^{(k-D)(s+nc)}\mathbb{E}\,C_{k}\big{(}F(e^{-(s+nc)})\big{)}=\frac{1}{\eta}\sum_{m=0}^{\infty}e^{(k-D)(s+mc)}R_{k,L}\big{(}e^{-(s+mc)}\big{)}.$

  * (III)

In general,

$\overline{C}_{k,F}^{\operatorname{frac}}:=\lim_{\delta\rightarrow
0}\frac{1}{|\ln\delta|}\int_{\delta}^{1}\varepsilon^{D-k}\mathbb{E}\,C_{k}(F(\varepsilon))~{}\varepsilon^{-1}d\varepsilon=\frac{1}{\eta}\int_{0}^{L}r^{D-k-1}R_{k,L}(r)\,dr.$

######  Remark 3.3.

In conditions (ii) and (iii) the boundary signs $\partial$ can be omitted,
since $\operatorname{int}F_{\sigma}(r)\subset\operatorname{int}F(r)$ for any
$\sigma\in\Sigma_{*}$ and the curvature measures are concentrated on the
boundary of the set $F(r)$. Similarly as in the deterministic case (see [Wi11,
Example 4.10]) one can construct an example of a homogeneous random fractal
where these conditions are not satisfied.

######  Remark 3.4.

It is a consequence of the statement that the limit expressions in (I), (II)
and (III) do not depend on the choice of the constant $L$, which gives some
flexibility in applications, see Section 5. This independence can also be seen
directly using the self-similarity of $F$, cf. Lemma 4.5. The proof of Theorem
3.2 will be given for the choice $L=R$, where $R$ is the constant appearing in
the conditions (ii) and (iii). It is easily seen that, if these two conditions
are satisfied with some $R>\sqrt{2}|O|$, then they are also satisfied with any
other constant $\tilde{R}>\sqrt{2}|O|$ instead of $R$. (Indeed, this is
obvious for $\tilde{R}<R$, since in this case the suprema in (ii) are taken
over a smaller range of values and also the expectation in (iii) needs to be
bounded for a smaller range of values $r$ only. For $\tilde{R}>R$, this
follows from Lemma 2.1, which implies that, for almost all $R<r\leq\tilde{R}$,
almost surely

$\displaystyle
C_{k}^{\operatorname{var}}\bigg{(}F(r),\partial(F_{\sigma}(r))\cap\partial\big{(}\bigcup_{\sigma^{\prime}\in\Sigma(r),~{}\sigma^{\prime}\neq\sigma}F_{\sigma^{\prime}}(r)\big{)}\bigg{)}\leq
C_{k}^{\operatorname{var}}\big{(}F(r)\big{)}\leq c_{k}(R)\tilde{R}^{k}.$

Thus, if conditions (ii) and (iii) hold with $R$, they also hold with
$\tilde{R}>R$, the latter possibly with a larger constant $C$.) This shows
that a proof of Theorem 3.2 for some $L>\sqrt{2}|O|$ (e.g. for $L=R$) implies
indeed the validity of the stated formulas for any $L>\sqrt{2}|O|$. The
argument in Lemma 4.5 below shows that the latter restriction can be relaxed
to $L>0$.

######  Remark 3.5.

The special case of deterministic self-similar sets satisfying OSC is included
in Theorem 3.2. In this case, statement and formulas reduce to the ones
obtained in [Wi08, Za11] (and in [Ga00] for the case $k=d$). The formulas in
[Wi08, Thm. 2.3.6] are stated with $L=1$ and the ones in [Za11, Thm. 2.3.8]
correspond to the choice $L=R$, but Remark 3.4 applies similarly in these
situations.

Note also that the formulas for the mean fractal curvatures are structurally
equal to the ones obtained for the fractal curvatures (almost surely and in
the mean) for self-similar random sets in [Ga00, Za11]. Indeed, if $F$ is a
homogeneous random fractal as in Theorem 3.2 generated by some random IFS, and
$K$ denotes the random self-similar set generated by the same random IFS, then
the formulas for their mean fractal curvatures coincide, except that in the
integrand $R_{k,L}$ the set $F$ has to be replaced by $K$. It turns out that
in certain situations, both functions coincide, see Example 5.3. In general,
this is probably not true.

##  4\. Proofs

The proof of Theorem 3.2 is split into several steps. We start with the main
part in which the problem is reduced to an application of the classical
Renewal theorem. In order to verify the assumptions of this theorem, we will
show that the function arising in the renewal equation is Lebesgue-a.e.
continuous and bounded by a directly Riemann integrable function. For the
first property condition (ii) is used and for the second one condition (iii).
The required estimates are shown in a sequence of lemmas. Part of the
estimates can be reduced to Lemma 2.1 (see Lemma 4.2 below). The others follow
from Lemmas 4.3 and 4.4, where the latter is essential. Finally, in Lemma 4.5
it is shown that the limit formula in the assertion of the theorem is the same
for all $L>0$.

###### Proof of Theorem 3.2 for $L=R$ and up to the estimates (4.1) and (4.4).

For
$L>0$ the function $\xi_{k}^{L}$ is $(\mathbb{P}\times\mathcal{L})$-almost
everywhere defined by

$\xi_{k}^{L}(r)=\xi_{k}^{L}(\omega,r):=\mathbf{1}_{(0,L]}(r)C_{k}(F(r))-\sum_{i=1}^{N}\mathbf{1}_{(0,Lr_{i}]}(r)C_{k}(F_{i}(r)),\quad(\omega,r)\in\operatorname{Reg}_{*}.$

Below we will see that the expectations of the absolute values of the two
summands on the right hand side are finite. Then in view of (3.14) the
function

$R_{k,L}=\mathbb{E}\xi_{k}^{L}\,$

in Theorem 3.2 is determined at a.a. arguments. If $L=R$, where $R$ is as in
Theorem 3.2, we will omit the subscript $R$ and write
$\xi_{k}(r):=\xi_{k}^{R}(r)$ and

$R_{k}(r)=\mathbb{E}\xi_{k}(r)\,.$

By the motion invariance and scaling property of $C_{k}$, we get for
$(\omega,r)$ as above,

$\xi_{k}(r)=\mathbf{1}_{(0,R]}(r)C_{k}(F(r))-\sum_{i=1}^{N}\mathbf{1}_{(0,R]}(r/r_{i})r_{i}^{k}C_{k}(F^{(1)}(r/r_{i}))\,.$

In order to translate the problem into the language of the Renewal theorem, we
substitute $r=Re^{-t}$ and define

$\displaystyle Z_{k}(t)=Z_{k}(\omega,t)$
$\displaystyle:=\mathbf{1}_{[0,\infty)}(t)e^{(k-D)t}C_{k}(F(Re^{-t})),$
$\displaystyle z_{k}(t)=z_{k}(\omega,t)$
$\displaystyle:=e^{(k-D)t}\xi_{k}(Re^{-t}),$

whenever $(\omega,Re^{-t})\in\operatorname{Reg}_{*}$. Note that
$z_{k}(\omega,t)=0$ for $t<0$. We infer from the above relations that for such
$(\omega,t)$,

$Z_{k}(\omega,t)=\sum_{i=1}^{N}r_{i}^{D}Z_{k}(\theta(\omega),t-|\ln
r_{i}|)+z_{k}(\omega,t),$

where $(\theta(\omega),t-|\ln r_{i}(\omega)|)\in\operatorname{Reg}_{*}$ for
$i=1,\ldots,N(\omega)$.

Denote

$T:=\left\\{t>0:\left(\omega,Re^{-t}\right)\in\operatorname{Reg}_{*}~{}{\rm
for}~{}\mathbb{P}\text{- a.a.}~{}\omega\right\\}.$

Let $\mu^{\ast n}$ be the $n$th convolution power of the distribution
$\mu=\mathbb{E}\sum_{i=1}^{N}\mathbf{1}_{(\cdot)}(|\ln r_{i}|)r_{i}^{D}$, if
$n\geq 1$, $\mu^{\ast 0}$ the Dirac measure at $0$, and

$U(t):=\sum_{n=0}^{\infty}\mu^{\ast n}((0,t]),~{}t>0.$

Note that the summands on the right vanish for $n>t/|\ln r_{\max}|$, so that
the summation is finite for each $t>0$. (In the sequel $U$-a.a. means a.a.
with respect to the corresponding measure.) Below we will show the following.

######  Lemma 4.1.

$t\in T$ implies $t-s\in T$ for $\mu$-a.a. $s\leq t$ and for $U$-a.a. $s\leq
t$.

Then we infer for $t\in T$ from the above equality for $Z_{k}(\omega,t)$ that

$\displaystyle\mathbb{E}|Z_{k}(t)|\leq$
$\displaystyle\int\sum_{i=1}^{N(\omega)}r_{i}(\omega)^{D}\left|Z_{k}\left(\theta(\omega),t-|\ln
r_{i}(\omega)|\right)\right|\mathbb{P}(d\omega)+\mathbb{E}|z_{k}(t)|$
$\displaystyle=$
$\displaystyle\int\mathbb{E}|Z_{k}(t-s)|\mu(ds)+\mathbb{E}|z_{k}(t)|\,.$

In view of Lemma 4.1 we obtain from iterated application of this inequality
that

$\mathbb{E}|Z_{k}(t)|\leq\int_{0}^{t}\mathbb{E}|z_{k}(t-s)|dU(s),\quad t\in
T.$

Below we will show that there exist some constants $c_{k}>0$ and $\delta>0$
such that for all $u\in T$ we have

(4.1) $\mathbb{E}|z_{k}(u)|\leq c_{k}\mathbf{1}_{[0,\infty)}(u)e^{-\delta u}.$

(In the proof for $k\leq d-2$ we will use condition (iii) of Theorem 3.2.)
Furthermore, since in our case $U(t)\leq t/|\ln r_{\max}|$ for all $t>0$, we
infer from (4.1) and the definition of $Z_{k}$ that for some constants $d_{k}$
and $d^{\prime}_{k}$,

$\mathbb{E}|Z_{k}(t)|\leq\int_{0}^{t}\mathbb{E}|z_{k}(t-s)|dU(s)<d_{k}\,,~{}t\in
T,$

and consequently,

(4.2) $\mathbb{E}|C_{k}(F(r))|\leq
d^{\prime}_{k}\,r^{k-D}|\ln(r/R)|~{},~{}~{}\mbox{for a.a.}~{}0<r\leq R.$

This shows in particular the finiteness of the expectations mentioned at the
beginning of the proof.

Moreover, we can repeat to above arguments omitting the absolute value signs
and replacing the corresponding inequalities by equalities in order to obtain
the renewal equation in the sense of Feller [Fe71]. We get for all $t\in T$

$\mathbb{E}Z_{k}(t)=\int_{0}^{t}\mathbb{E}Z_{k}(t-s)\mu(ds)+\mathbb{E}z_{k}(t)$

and

(4.3)
$\displaystyle\mathbb{E}Z_{k}(t)=\int_{0}^{t}\mathbb{E}z_{k}(t-s)\,dU(s).$

Below we will also show that for all $0<r_{0}\leq R$,

(4.4) $\mathbb{E}\operatorname{ess\,sup}_{r>r_{0}}\limits|\xi_{k}(r)|<\infty,$

where $\operatorname{ess\,sup}$ means the supremum over all arguments where
the function is determined. (In the proof of (4.4) for $k\leq d-2$ we will use
condition (ii) of the theorem.)

We now define two auxiliary functions on $(0,\infty)$ by

$\displaystyle\overline{z}_{k}(t)$
$\displaystyle:=\begin{cases}\mathbb{E}z_{k}(t),&{\rm if}~{}t\in T,\\\
\limsup\limits_{\begin{subarray}{c}t^{\prime}\to t\\\ t^{\prime}\in
T\end{subarray}}\mathbb{E}z_{k}(t^{\prime}),&{\rm
if}~{}t\in(0,\infty)\setminus T,\end{cases}$
$\displaystyle\overline{Z}_{k}(t)$
$\displaystyle:=\int_{0}^{t}\overline{z}_{k}(t-s)dU(s),~{}t>0.$

Then in view of Lemma 4.1 and (4.3),

(4.5) $\displaystyle\overline{Z}_{k}(t)=\mathbb{E}Z_{k}(t)\quad\text{ for
}t\in T.$

Assumption (i) and Lemma 2.4 imply that the random function
$\xi_{k}(\omega,\cdot)$ is continuous in the second argument at all arguments
$r$, where $(\omega,r)\in\operatorname{Reg}_{*}$. (For $k=d,d-1$ we do not
need (i) for this conclusion.) The dominated convergence theorem (justified by
(4.4)) yields that at each $t\in T$ the function $\overline{z}_{k}$, which
agrees with $e^{(k-D)t}\mathbb{E}\xi_{k}(Re^{-t})$ at such $t$, is continuous,
i.e., $\overline{z}_{k}$ is continuous Lebesgue-a.e.. Moreover, in view of
(4.1), $\overline{z}_{k}$ is bounded by a directly Riemann integrable
function. Thus, according to Asmussen [As87, Prop. 4.1, p. 118],
$\overline{z}_{k}$ is directly Riemann integrable, too. Therefore the
classical Renewal theorem in Feller [Fe71, p. 363] can be applied, which
yields that

$\lim_{t\rightarrow\infty}\overline{Z}_{k}(t)=\frac{1}{\eta}\int_{0}^{\infty}\overline{z}_{k}(t)\,dt.$

The right hand side agrees with

$\frac{1}{\eta}\int_{0}^{\infty}\mathbb{E}z_{k}(t)\,dt=\frac{1}{\eta}\int_{0}^{\infty}e^{(k-D)t}R_{k}(Re^{-t})\,dt,$

since $\overline{z}_{k}(t)=\mathbb{E}z_{k}(t)$ for Lebesgue-a.a. $t$.
Multiplying $R^{D-k}$ in this equation and substituting $r=Re^{-t}$ under the
integral, assertion (I) follows in view of (4.5).

Since $\mathbb{E}Z_{k}$ is bounded on finite intervals, in the non-lattice
case the corresponding average limit in (III) is a consequence.

In the lattice case, the Renewal theorem provides the limit along arithmetic
progressions with respect to the lattice constant, here only for those
sequences along which the function is determined. This shows assertion (II).
This also implies the average convergence (III). For more details we refer to
the arguments of Gatzouras at the end of the proof of [Ga00, Theorem 2.3] in
the classical case. ∎

###### Proof of Lemma 4.1.

For $t\in T$ we have

$\displaystyle 1$
$\displaystyle=\mathbb{P}\left(\left\\{\omega:(Re^{-t},F_{\sigma}(\omega))\in\operatorname{Reg}~{}\mbox{for
all}~{}\sigma\in\Sigma_{*}(\omega)\right\\}\right)$
$\displaystyle\leq\mathbb{P}\left(\left\\{\omega:(Re^{-t},F_{i\tau}(\omega))\in\operatorname{Reg}\text{
for
}i=1,\ldots,N(\omega)\,,~{}\tau\in\Sigma_{*}(\theta(\omega))\right\\}\right).$

Using that $(Re^{-t},F_{i\tau}(\omega))\in\operatorname{Reg}$ if and only if
$(Re^{-(t-|\ln
r_{i}(\omega)|)},F_{\tau}(\theta(\omega)))\in\operatorname{Reg}$ and the
product structure of the basis probability space, we infer

$\displaystyle 1=\int\mathbb{P}\Big{(}\Big{\\{}\omega^{\prime}:\big{(}R$
$\displaystyle e^{-(t-|\ln
r_{i}(\omega)|)},F_{\tau}(\omega^{\prime})\big{)}\in\operatorname{Reg}\text{
for }$ $\displaystyle\qquad\quad
i=1,\ldots,N(\omega),\tau\in\Sigma_{*}(\omega^{\prime})\Big{\\}}\Big{)}\mathbb{P}(d\omega)\,.$

Hence, we get for $\mathbb{P}$-a.a. $\omega$,

$\mathbb{P}\left(\left\\{\omega^{\prime}:\left(Re^{-(t-|\ln
r_{i}(\omega)|)},F_{\tau}(\omega^{\prime})\right)\in\operatorname{Reg}\text{
for }\tau\in\Sigma_{*}(\omega^{\prime})\right\\}\right)=1\,,$

for $i=1,\ldots,N(\omega)$, and therefore

$\int\sum_{i=1}^{N}r_{i}(\omega)^{D}\mathbb{P}\left(\left\\{\omega^{\prime}:\left(Re^{-(t-|\ln
r_{i}(\omega)|)},F_{\tau}(\omega^{\prime})\right)\in\operatorname{Reg}\text{
for }\tau\in\Sigma_{*}(\omega^{\prime})\right\\}\right)\mathbb{P}(d\omega)=1,$

since $\int\sum_{i=1}^{N}r_{i}(\omega)^{D}\mathbb{P}(d\omega)=1$. By the
definition of the measure $\mu$, this means that

$\int\mathbb{P}\left(\left\\{\omega^{\prime}:\left(Re^{-(t-s)},F_{\tau}(\omega^{\prime})\right)\in\operatorname{Reg}\text{
for }\tau\in\Sigma_{*}(\omega^{\prime})\right\\}\right)\mu(ds)=1\,.$

Thus, for $\mu$-a.a. $s\leq t$,

$\mathbb{P}\left(\left\\{\omega^{\prime}:\left(Re^{-(t-s)},F_{\tau}(\omega^{\prime})\right)\in\operatorname{Reg}\text{
for }\tau\in\Sigma_{*}(\omega^{\prime})\right\\}\right)=1,$

i.e., for $t\in T$ and $\mu$-a.a. $s\leq t$ we get $t-s\in T$. Iterated
application of this result yields that $t\in T$ implies $t-s\in T$ for
$U$-a.a. $s\leq t$. ∎

In order to complete the proof of Theorem 3.2, it remains to verify (4.1) and
(4.4). By the definition of $z_{k}$, the first one is equivalent to showing
that there are constants $c_{k}^{\prime}$ and $\delta>0$ such that

(4.6) $\mathbb{E}\big{|}\xi_{k}(r)\big{|}\leq
c_{k}^{\prime}\,r^{k-D+\delta}~{}~{}\mbox{for a.a.}~{}r>0\,.$

(In particular, if this inequality holds, then (4.1) is satisfied with the
same $\delta$ and $c_{k}=R^{k-D+\delta}c_{k}^{\prime}$.) Observe that, by
Lemma 2.1, (4.4) and (4.6) are satisfied for all $r_{0}>R$ and $r>R$,
respectively. In order to prepare the estimates for a.a. $r\leq R$ note first
that for a.a. $r\in(0,R]$ we get

$\displaystyle|\xi_{k}(r)|$
$\displaystyle=\bigg{|}C_{k}(F(r))-\sum_{i=1}^{N}(1-\mathbf{1}_{(Rr_{i},R]}(r))C_{k}(F_{i}(r))\bigg{|}$
$\displaystyle=\bigg{|}C_{k}(F(r))-\sum_{i=1}^{N}C_{k}(F_{i}(r))+\sum_{i=1}^{N}\mathbf{1}_{(Rr_{i},R]}(r)C_{k}(F_{i}(r))\bigg{|}$
$\displaystyle\leq\bigg{|}C_{k}(F(r))-\sum_{i=1}^{N}C_{k}(F_{i}(r))\bigg{|}+\bigg{|}\sum_{i=1}^{N}\mathbf{1}_{(Rr_{i},R]}(r)C_{k}(F_{i}(r))\bigg{|}$
$\displaystyle=:\xi_{k1}(r)+\xi_{k2}(r)\,.$

Therefore, it suffices to provide the required bounds for the two summands in
the last line instead of $|\xi_{k}(r)|$. For the second summand the estimates
(4.4) and (4.6) follow from the next statement, since we supposed that
$\mathbb{E}N<\infty$.

######  Lemma 4.2.

There exists some constant $b_{k}$ such that for all $r\in(0,R]$,

$\xi_{k2}(r)\leq b_{k}Nr^{k},~{}a.s.\,.$

###### Proof.

Recalling that $F_{i}(r)=(f_{i}(F^{(1)}))(r)=f_{i}(F^{(1)}(r/r_{i}))$, we have

$\displaystyle\xi_{k2}(r)=\left|\sum_{i=1}^{N}\mathbf{1}_{(Rr_{i},R]}(r)C_{k}(F_{i}(r))\right|$
$\displaystyle\leq\left|\sum_{i=1}^{N}\mathbf{1}_{(Rr_{i},R]}(r)C_{k}^{\operatorname{var}}(f_{i}(F^{(1)}(r/r_{i})))\right|$
$\displaystyle=\sum_{i=1}^{N}\mathbf{1}_{(Rr_{i},R]}(r)r_{i}^{k}C_{k}^{\operatorname{var}}(F^{(1)}(r/r_{i})).$

By the choice of $R$, we have for any $r\in(Rr_{i},R]$ that
$r/r_{i}>R>\sqrt{2}|O|\geq\sqrt{2}|F^{(1)}|$. Hence, we can apply Lemma 2.1
and infer that there is a constant $b_{k}=b_{k}(R)$ such that almost surely

$C_{k}^{\operatorname{var}}(F^{(1)}(r/r_{i}))\leq b_{k}(r/r_{i})^{k}$

for all $r>Rr_{i}$ and all $i$. Plugging this into the above estimates we get
for all $r\leq R$,

$\left|\sum_{i=1}^{N}\mathbf{1}_{(Rr_{i},R]}(r)C_{k}(F_{i}(r))\right|\leq
Nb_{k}r^{k}\,,$

i.e., the assertion. ∎

It remains to prove the required bounds for the first summand $\xi_{k1}$ in
the above estimates.

######  Lemma 4.3.

There exist constants $c_{k}^{\prime}>0$ and $\delta>0$ such that

(4.7)
$\mathbb{E}\xi_{k1}(r)=\mathbb{E}\big{|}C_{k}(F(r))-\sum_{i=1}^{N}C_{k}(F_{i}(r))\big{|}\leq
c_{k}^{\prime}\,r^{k-D+\delta}\,,~{}\mbox{for a.a.}~{}r\in(0,R],$

and

(4.8) $\mathbb{E}\operatorname{ess\,sup}_{r_{0}<r\leq
R}\limits\xi_{k1}(r)<\infty,\quad 0<r_{0}<R.$

###### Proof.

For $k\in\\{d-1,d\\}$ the estimate (4.8) is a simple consequence of the facts
that for any compact $K$ and $r>0$ we have
$C_{d}(K(r))\leq\operatorname{const}(2r+|K|)^{d}$ and
$C_{d-1}(K(r))\leq\operatorname{const}r^{-1}C_{d}(K(r))$. The latter follows
from the Kneser property of the volume function, i.e.,
$\frac{\partial}{\partial
r}\mathcal{L}^{d}(K(r))\leq\frac{d}{r}\mathcal{L}^{d}(K(r))$ for almost all
$r$ (see e.g.[RSS09, Lemma 4.6 and its proof]), together with [RW10, Corollary
2.6].

For $k\leq d-2$ we decompose the total $k$th curvatures by means of the
corresponding curvature measures:

$C_{k}(F(r))=C_{k}(F(r),A_{r})+C_{k}\big{(}F(r),(A_{r})^{c}\big{)},$

where

(4.9) $A_{r}:=\bigcup_{j\neq k}f_{j}(\overline{O})(r)\cap
f_{k}(\overline{O})(r)\,.$

Similarly,

$C_{k}(F_{i}(r))=C_{k}(F_{i}(r),A_{r})+C_{k}\big{(}F_{i}(r),(A_{r})^{c}\big{)}\,,\quad
i=1,\ldots,N\,.$

The local definiteness of the curvature measure $C_{k}$ implies

$C_{k}(F_{i}(r),(A_{r})^{c})=C_{k}(F_{i}(r),B^{i})=C_{k}(F(r),B^{i})$

and $F(r)\cap(A_{r})^{c}$ is the disjoint union of the sets
$B^{i}:=F_{i}(r)\setminus A_{r}$, $i=1,\ldots,N$. Hence,

$C_{k}(F(r),(A_{r})^{c})-\sum_{i=1}^{N}C_{k}(F_{i}(r),(A_{r})^{c})=0\,.$

Substituting this in the above expression for $\xi_{k1}$, we infer that

$\xi_{k1}(r)=\big{|}C_{k}(F(r),A_{r})-\sum_{i=1}^{N}C_{k}(F_{i}(r),A_{r})\big{|}$

and so, by the scaling property of $C_{k}$,

$\displaystyle\xi_{k1}(r)$ $\displaystyle=$
$\displaystyle\bigg{|}C_{k}(F(r),A_{r})-\sum_{i=1}^{N}r^{k}_{i}\,C_{k}\big{(}F^{(1)}\big{(}\frac{r}{r_{i}}\big{)},f_{i}^{-1}(A_{r})\big{)}\bigg{|}$
$\displaystyle=$
$\displaystyle\big{|}\xi_{k3}(r)-\xi_{k4}(r)-\xi_{k5}(r)\big{|}\leq\left|\xi_{k3}(r)\right|+\left|\xi_{k4}(r)\right|+\left|\xi_{k5}(r)\right|$

where

$\displaystyle\xi_{k3}(r)$ $\displaystyle:=$ $\displaystyle
C_{k}(F(r),A_{r}),$ $\displaystyle\xi_{k4}(r)$ $\displaystyle:=$
$\displaystyle\sum_{i=1}^{N}r_{i}^{k}\,\mathbf{1}_{(0,R]}\big{(}\frac{r}{r_{i}}\big{)}\,C_{k}\bigg{(}F^{(1)}\big{(}\frac{r}{r_{i}}\big{)},f_{i}^{-1}(A_{r})\bigg{)},$
$\displaystyle\xi_{k5}(r)$ $\displaystyle:=$
$\displaystyle\sum_{i=1}^{N}r_{i}^{k}\,\mathbf{1}_{(R,\infty)}\big{(}\frac{r}{r_{i}}\big{)}\,C_{k}\bigg{(}F^{(1)}\big{(}\frac{r}{r_{i}}\big{)},f_{i}^{-1}(A_{r})\bigg{)}\,.$

Therefore, instead of proving the estimates (4.7) and (4.8) for $\xi_{k1}$, it
suffices to prove corresponding estimates for $|\xi_{k3}|$, $|\xi_{k4}|$ and
$|\xi_{k5}|$ separately.

The arguments for $|\xi_{k5}|$ are the same as for $\xi_{k2}$ taking into
account that for any Borel set $B$, $|C_{k}(F^{(1)}(r),B)|\leq
C_{k}^{\operatorname{var}}(F^{(1)}(r),\mathbb{R}^{d})$ and applying Lemma 2.1.

For estimating $\xi_{k3}$ and $\xi_{k4}$ we will use the set inclusions

$A_{r}\subset(f(O))^{c}(r)\,,\quad f_{i}^{-1}(A_{r})\subset
O^{c}\big{(}\frac{r}{r_{i}})\,,\quad{\rm and}~{}~{}O^{c}(r)\subset
f(O)^{c}(r)\,.$

(Recall that $f(O)=\bigcup_{i=1}^{N}f_{i}(O)$ and $O$ is the open set from
UOSC.) Then for $|\xi_{k3}|$ the estimates (4.7) and (4.8) follow from (4.11)
and (4.10), respectively, in Lemma 4.4 below.

Furthermore, for Lebesgue-a.a. $r\in(r_{0},R]$, we obtain from the above set
inclusions

$\displaystyle|\xi_{k4}(r)|$ $\displaystyle\leq$
$\displaystyle\sum_{i=1}^{N}\mathbf{1}_{(0,R]}(\frac{r}{r_{i}})r_{i}^{k}C_{k}^{\operatorname{var}}\big{(}F^{(1)}(\frac{r}{r_{i}}),O^{c}(\frac{r}{r_{i}})\big{)}$
$\displaystyle\leq$ $\displaystyle N\operatorname{ess\,sup}_{r_{0}<r\leq
R}\limits C_{k}^{\operatorname{var}}\big{(}F^{(1)}(r),O^{c}(r)\big{)}\,.$

Since the random set $F^{(1)}$ is independent of the events in the first step,
in particular of $N$, and has the same distribution as $F$ we infer

$\mathbb{E}\operatorname{ess\,sup}_{r_{0}<r\leq
R}\limits|\xi_{k4}(r)|\leq\mathbb{E}N\,\mathbb{E}\operatorname{ess\,sup}_{r_{0}<r\leq
R}\limits C_{k}^{\operatorname{var}}\big{(}F(r),O^{c}(r)\big{)}\,.$

As $O^{c}(r)\subset f(O)^{c}(r)$, the estimate (4.10) in Lemma 4.4 below
yields (4.8) for $\xi_{k4}$.

Similarly we get

$\displaystyle\operatorname{ess\,sup}_{r_{0}<r\leq R}\limits$
$\displaystyle\mathbb{E}\left[\left(\frac{|\xi_{k4}(r)|}{r^{k-D+\delta}}\right)\right]$
$\displaystyle\leq\operatorname{ess\,sup}_{r_{0}<r\leq
R}\limits\mathbb{E}\left[\sum_{i=1}^{N}r_{i}^{D-\delta}\mathbf{1}_{(0,R]}(\frac{r}{r_{i}})\left(\frac{r_{i}^{k-D+\delta}}{r^{k-D+\delta}}~{}C_{k}^{\operatorname{var}}\bigg{(}F^{(1)}\big{(}\frac{r}{r_{i}}\big{)},O^{c}\big{(}\frac{r}{r_{i}}\big{)}\bigg{)}\right)\right]$
$\displaystyle=\operatorname{ess\,sup}_{r_{0}<r\leq
R}\limits\mathbb{E}\left[\sum_{i=1}^{N}r_{i}^{D-\delta}\mathbf{1}_{(0,R]}(\frac{r}{r_{i}})\left(\frac{r_{i}^{k-D+\delta}}{r^{k-D+\delta}}~{}\mathbb{E}C_{k}^{\operatorname{var}}\bigg{(}F^{(1)}\big{(}\frac{r}{r_{i}}\big{)},O^{c}\big{(}\frac{r}{r_{i}}\big{)}\bigg{)}\right)\right]$
$\displaystyle\leq\mathbb{E}\left(\sum_{i=1}^{N}r_{i}^{D-\delta}\right)\operatorname{ess\,sup}_{0<r\leq
R}\limits\left(\frac{\mathbb{E}C_{k}^{\operatorname{var}}\big{(}F^{(1)}(r),O^{c}(r)\big{)}}{r^{k-D+\delta}}\right)$
$\displaystyle\leq\mathbb{E}N~{}\operatorname{ess\,sup}_{0<r\leq
R}\limits\left(\frac{\mathbb{E}C_{k}^{\operatorname{var}}\big{(}F(r),f(O)^{c}(r)\big{)}}{r^{k-D+\delta}}\right)<\infty$

for some $0<\delta<D$ according to (4.11) in Lemma 4.4 below. (Via conditional
expectation the inner expectations are chosen with respect to the random set
$F^{(1)}$. Recall that the latter is independent of the contraction ratios
$r_{1},\ldots,r_{N}$ and has the same distribution as $F$.) This yields the
remaining estimate (4.7) for $\xi_{k4}$. Thus, Lemma 4.4 completes the proof
of Theorem 3.2. ∎

######  Lemma 4.4.

Under the conditions of Theorem 3.2 we have

(4.10) $\mathbb{E}\operatorname{ess\,sup}_{r_{0}<r<R}\limits
C_{k}^{\operatorname{var}}\big{(}F(r),f(O)^{c}(r)\big{)}<\infty$

for all $0<r_{0}<R$, and

(4.11)
$\operatorname{ess\,sup}_{0<r<R}\limits\left(\frac{\mathbb{E}C_{k}^{{\operatorname{var}}}\big{(}F(r),f(O)^{c}(r)\big{)}}{r^{k-D+\delta}}\right)<\infty$

for some $0<\delta<D$.

###### Proof.

We start similarly as in the above proof choosing the subtree $\Sigma(r)$ from
(3.6) instead of $\Sigma_{1}$ in the decomposition of the curvature measures.
First note that by a simple volume comparing argument the Uniform Open Set
Condition (3.3) implies that for all $r>r_{0}>0$,

(4.12) $\sharp\big{(}\Sigma(r)\big{)}\leq\operatorname{const}(r_{0})^{-d}\,,$

where $\sharp$ denotes the number of elements of a finite set.

Next recall that $F(r)=\bigcup_{\sigma\in\Sigma(r)}F_{\sigma}(r)$ for any
$r>0$. Since curvature measures are locally defined, this implies

(4.13) $\displaystyle
C_{k}^{\operatorname{var}}\big{(}F(r),f(O)^{c}(r)\big{)}$ $\displaystyle=$
$\displaystyle
C_{k}^{\operatorname{var}}\bigg{(}F(r),\big{(}\bigcup_{\sigma\in\Sigma(r)}F_{\sigma}(r)\big{)}\cap
f(O)^{c}(r)\bigg{)}$ $\displaystyle\leq$
$\displaystyle\sum_{\sigma\in\Sigma_{b}(r)}C_{k}^{\operatorname{var}}(F(r),F_{\sigma}(r))\,,$

where the boundary code set $\Sigma_{b}(r)$ was defined in (3.8). Note that
$\operatorname{int}F_{\sigma}(r)\subset\operatorname{int}F(r)$ for any
$\sigma\in\Sigma(r)$, thus
$C_{k}^{\operatorname{var}}(F(r),\operatorname{int}F_{\sigma}(r))=0$ for
$k\leq d-1$ (since the curvature measures are concentrated on the boundary of
$F(r)$). Then for $k\in\\{d-1,d\\}$ we can use the (in)equality

(4.14) $C_{k}^{\operatorname{var}}\big{(}F(r),F_{\sigma}(r)\big{)}\leq
C_{k}\big{(}F_{\sigma}(r)\big{)}\,,~{}~{}\sigma\in\Sigma_{b}(r)\,.$

Similarly we obtain for any $\sigma\in\Sigma_{b}(r)$ and $k\leq d-2$,

$\displaystyle C_{k}^{\operatorname{var}}(F(r),F_{\sigma}(r))$
$\displaystyle=C_{k}^{\operatorname{var}}\bigg{(}F(r),F_{\sigma}(r)\setminus\bigcup_{\sigma^{\prime}\in\Sigma(r),\,\sigma^{\prime}\neq\sigma}F_{\sigma^{\prime}}(r)\bigg{)}+C_{k}^{\operatorname{var}}\bigg{(}F(r),F_{\sigma}(r)\cap\bigcup_{\sigma^{\prime}\in\Sigma(r),\,\sigma^{\prime}\neq\sigma}F_{\sigma^{\prime}}(r)\bigg{)}$
$\displaystyle\leq
C_{k}^{\operatorname{var}}(F_{\sigma}(r))+C_{k}^{\operatorname{var}}\bigg{(}F(r),\partial
F_{\sigma}(r)\cap\partial\big{(}\bigcup_{\sigma^{\prime}\in\Sigma(r),\,\sigma^{\prime}\neq\sigma}F_{\sigma^{\prime}}(r)\big{)}\bigg{)}$
$\displaystyle=r^{k}_{\sigma}\,C_{k}^{\operatorname{var}}\bigg{(}F^{(|\sigma|)}\big{(}\frac{r}{r_{\sigma}}\big{)}\bigg{)}+C_{k}^{\operatorname{var}}\bigg{(}F(r),\partial
F_{\sigma}(r)\cap\partial\big{(}\bigcup_{\sigma^{\prime}\in\Sigma(r),\,\sigma^{\prime}\neq\sigma}F_{\sigma^{\prime}}(r)\big{)}\bigg{)}$
$\displaystyle=:S_{1}(r,\sigma)+S_{2}(r,\sigma)\,.$

Moreover, for $\sigma\in\Sigma_{b}(r)$ we have $\frac{r}{r_{\sigma}}\geq R$.
Hence, the first summand $S_{1}(r,\sigma)$ on the right hand side is bounded
by a constant in view of Lemma 2.1, since $|O|\geq|F^{(|\sigma|)}|$. (This
holds also for $k\in\\{d-1,d\\}$.) Therefore, these estimates together with
(4.13) and (4.12) lead to

$\displaystyle\mathbb{E}\operatorname{ess\,sup}_{r_{0}<r\leq R}\limits$
$\displaystyle
C_{k}^{\operatorname{var}}\big{(}F(r),f(\overline{O})^{c}(r)\big{)}$
$\displaystyle\leq\mathbb{E}\operatorname{ess\,sup}_{r_{0}<r\leq
R}\limits\sum_{\sigma\in\Sigma_{b}(r)}S_{1}(r,\sigma)+\mathbb{E}\operatorname{ess\,sup}_{r_{0}<r\leq
R}\limits\sum_{\sigma\in\Sigma_{b}(r)}S_{2}(r,\sigma)$
$\displaystyle\leq\operatorname{const}+\operatorname{const}\mathbb{E}\operatorname{ess\,sup}_{r_{0}<r\leq
R}\limits\max_{\sigma\in\Sigma_{b}(r)}S_{2}(r,\sigma)\,.$

In the last estimate we have used that $\sharp(\Sigma_{b}(r))$ is bounded by a
constant (depending on $r_{0}$) according to (4.12). Since the last summand is
finite by assumption (ii) in our theorem, we obtain the first assertion
(4.10).

To prove (4.11), we argue similarly that for $0<\delta<D$,

$\displaystyle
r^{D-\delta-k}\mathbb{E}C_{k}^{\operatorname{var}}\big{(}F(r),f(O)^{c}(r)\big{)}$
$\displaystyle\quad\leq\mathbb{E}\bigg{(}r^{D-\delta}\sum_{\sigma\in\Sigma_{b}(r)}r^{-k}S_{1}(r,\sigma)\bigg{)}+\mathbb{E}\bigg{(}r^{D-\delta}\sum_{\sigma\in\Sigma_{b}(r)}r^{-k}S_{2}(r,\sigma)\bigg{)}$
$\displaystyle\quad\leq\mathbb{E}\big{(}r^{D-\delta}\sharp(\Sigma_{b}(r))\max_{\sigma\in\Sigma_{b}(r)}r^{-k}S_{1}(r,\sigma)\big{)}+\mathbb{E}\big{(}r^{D-\delta}\sharp(\Sigma_{b}(r))\max_{\sigma\in\Sigma_{b}(r)}r^{-k}S_{2}(r,\sigma)\big{)}\,.$

Since $\frac{r}{r_{\sigma}}\geq R$ for $\sigma\in\Sigma_{b}(r)$ and any
$0<r\leq R$, by Lemma 2.1, the maximum in the first summand is uniformly
bounded by some constant $c=c(R)$ independent of $r$. Therefore, for all
$r\in(0,R]$, the last sum does not exceed

$\displaystyle c\,$
$\displaystyle\mathbb{E}\big{[}r^{D-\delta}\sharp(\Sigma_{b}(r))\big{]}+\mathbb{E}\big{[}r^{D-\delta}\sharp(\Sigma_{b}(r))\max_{\sigma\in\Sigma_{b}(r)}r^{-k}S_{2}(r,\sigma)\big{]}$
$\displaystyle=c\,\mathbb{E}\big{[}r^{D-\delta}\sharp(\Sigma_{b}(r))\big{]}+\mathbb{E}\bigg{[}r^{D-\delta}\sharp(\Sigma_{b}(r))\mathbb{E}\big{[}\max_{\sigma\in\Sigma_{b}(r)}r^{-k}S_{2}(r,\sigma)\big{)}\big{|}\sharp(\Sigma_{b}(r))\big{]}\bigg{]}$
$\displaystyle\leq
c\,\mathbb{E}\big{[}r^{D-\delta}\sharp(\Sigma_{b}(r))\big{]}+\mathbb{E}\bigg{[}r^{D-\delta}\sharp(\Sigma_{b}(r))\operatorname{ess\,sup}_{0<r\leq
R}\limits\mathbb{E}\big{[}\max_{\sigma\in\Sigma_{b}(r)}r^{-k}S_{2}(r,\sigma)\big{)}\big{|}\sharp(\Sigma_{b}(r))\big{]}\bigg{]}$
$\displaystyle\leq(c+C)\,\mathbb{E}\big{[}r^{D-\delta}\sharp(\Sigma_{b}(r))\big{]}\,,$

since, by condition (iii) in Theorem 3.2, the supremum of the conditional
expectations in the second summand is bounded by $C$. Therefore, it remains to
show that for some $0<\delta<D$,

(4.15) $\sup_{0<r\leq
R}\limits\mathbb{E}\big{[}r^{D-\delta}\sharp(\Sigma_{b}(r))\big{]}<\infty\,.$

This was already proved in [Za20]. For convenience of the reader we replicate
the arguments here. To this aim we will use USOSC, i.e., UOSC with $O$ such
that $\mathbb{P}(F\cap O\neq\emptyset)>0$, which implies that there exist some
constants $\alpha>0$ and $0<\rho<1$ such that

(4.16) $\mathbb{P}(\Sigma(\rho,\alpha)\neq\emptyset)>0~{}~{}{\rm
for}~{}~{}\Sigma(\rho,\alpha):=\\{\tau\in\Sigma(\rho):d(x,\partial
O)>\alpha\,,\,x\in f_{\tau}(\overline{O})\\}\,.$

Since $\Sigma(\rho)$ is a Markov stop, one infers from
$\mathbb{E}\sum_{i=1}^{N}r_{i}^{D}=1$ that
$\mathbb{E}\sum_{\sigma\in\Sigma(\rho)}r_{\sigma}^{D}=1$ (see e.g. [Za20,
Proposition 1]). Then let $\delta$ be determined by

(4.17)
$\mathbb{E}\bigg{(}\sum_{\tau\in\Sigma(\rho)\setminus\Sigma(\rho,\alpha)}r_{\tau}^{D-\delta}\bigg{)}=1\,.$

We next choose for all $r>0$,

(4.18) $r^{*}:=2R(\alpha r_{\min})^{-1}r\,.$

Then we get for $i=1,\ldots,N$ and $i\sigma\in\Sigma(r^{*})$ with
$\sigma=\tau\sigma^{\prime}$ for some $\tau\in\Sigma^{(1)}(\rho,\alpha)$ that

$f_{i\sigma}(F^{(|i\sigma|)})(r)\cap f(O)^{c}(r)=\emptyset\,.$

(To see this note that for any $x\in f_{i\sigma}(F^{|i\sigma|})(r)$ there
exists a $y\in f_{i\sigma}(F^{(|i\sigma|)})$ such that $|x-y|\leq r$.
Furthermore, $y\in f_{i\sigma}(F^{(|i\sigma|)})\subset f_{i}(F^{(1)})\subset
f_{i}(\overline{O})\subset f(\overline{O})$, and $d(y,\partial f(O))\geq
d(y,\partial f_{i\sigma}(O))>r_{i\sigma}\alpha>R^{-1}r^{*}r_{\min}\alpha=2r$.
Consequently, $d(x,\partial f(O)^{c})\geq d(y,\partial
f(O)^{c})-|x-y|>2r-r=r$, i.e., $x\notin(f(O)^{c})(r)$.)

From this we obtain

$\sharp(\Sigma_{b}(r))\leq\sum_{i=1}^{N}\sharp(\\{w\in\Sigma(r^{*}):w=i\sigma,\,\sigma\in\Xi^{(1)}(r^{*}/r_{i})\\})\,,$

where the random sets $\Xi(r)$, $r>0$, are defined as

$\Xi(r):=\Sigma(r)\setminus\\{\sigma\in\Sigma(r):\sigma=\tau\sigma^{\prime}~{}~{}\mbox{for
some}~{}~{}\tau\in\Sigma(\rho,\alpha)\\}\,.$

In these notations we get

$\displaystyle r^{D-\delta}\mathbb{E}\sharp(\Sigma_{b}(r))$
$\displaystyle\leq$ $\displaystyle
r^{D-\delta}\mathbb{E}\sum_{i=1}^{N}\sharp(\Xi^{(1)}(r^{*}/r_{i}))$
$\displaystyle=$ $\displaystyle(\alpha
r_{\min}/2)^{D-d}\mathbb{E}\sum_{i=1}^{N}r_{i}^{D-\delta}(r^{*}/r_{i})^{D-\delta}\sharp(\Xi^{(1)}(r^{*}/r_{i}))$
$\displaystyle=$ $\displaystyle(\alpha
r_{\min}/2)^{D-d}\mathbb{E}\sum_{i=1}^{N}r_{i}^{D-\delta}(r^{*}/r_{i})^{D-\delta}\mathbb{E}\sharp(\Xi^{(1)}(r^{*}/r_{i}))$
$\displaystyle=$
$\displaystyle\operatorname{const}\mathbb{E}\sum_{i=1}^{N}r_{i}^{D-\delta}\psi(r^{*}/r_{i})\,,$

where we have used that $\Xi^{(1)}$ is independent of the events in the first
step and has the same distribution as $\Xi$ and then the notation
$\psi(r):=r^{D-\delta}\mathbb{E}\sharp(\Xi(r))$. Now it suffices to show that
the function $\psi$ is bounded.

Similarly as above, using (4.16) and the definition of $\Xi(r)$ we infer for
sufficiently large $M$ and $r<\rho$,

$\displaystyle\psi(r)$ $\displaystyle=$
$\displaystyle\mathbb{E}\sum_{\tau\in\Sigma(\rho)\setminus\Sigma(\rho,\alpha)}r_{\tau}^{D-\delta}(r/r_{\tau})^{D-\delta}\sharp(\Xi^{(|\tau|)}(r/r_{\tau})$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{M}\mathbb{E}\sum_{\stackrel{{\scriptstyle\tau\in\Sigma(\rho)\setminus\Sigma(\rho,\alpha)}}{{|\tau|=n}}}r_{\tau}^{D-\delta}(r/r_{\tau})^{D-\delta}\sharp(\Xi^{(n)}(r/r_{\tau})$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{M}\mathbb{E}\sum_{\stackrel{{\scriptstyle\tau\in\Sigma(\rho)\setminus\Sigma(\rho,\alpha)}}{{|\tau|=n}}}r_{\tau}^{D-\delta}\psi(r/r_{\tau})=\mathbb{E}\sum_{\tau\in\Sigma(\rho)\setminus\Sigma(\rho,\alpha)}r_{\tau}^{D-\delta}\psi(r/r_{\tau})$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\sum_{\tau\in\Sigma(\rho)\setminus\Sigma(\rho,\alpha)}r_{\tau}^{D-\delta}\operatorname{ess\,sup}_{r^{\prime}\geq
r/\rho}\limits\psi(r^{\prime})=\operatorname{ess\,sup}_{r^{\prime}\geq
r/\rho}\limits\psi(r^{\prime})\,,$

where we have used that the random sets $\Xi^{(n)}(r)$ are independent of the
behavior of the system up to the step $n$ via conditional expectation, that
they have the same distribution as $\Xi(r)$, and then (4.17). Hence,
$\psi(r)\leq\operatorname{ess\,sup}_{r^{\prime}\geq
r/\rho}\limits\psi(r^{\prime})$ for any $r<\rho$ which implies

$\sup_{r\geq\rho^{k+1}}\psi(r)\leq\sup_{r\geq\rho^{k}}\psi(r)~{}~{}\mbox{for
all}~{}k\,.$

Since the function $\psi$ is bounded on any finite interval away from zero it
is bounded on $(0,R)$. This completes the proof of (4.15). ∎

We have proved Theorem 3.2 for the special case $L=R$ (and it was argued in
Remark 3.4 that this implies, it holds for any $L>\sqrt{2}|O|$.) In order to
complete the proof, we need to show that the assertions hold also for
$L\in(0,\sqrt{2}|O|]$. Recall that $R_{k}=R_{k,R}$.

######  Lemma 4.5.

Under the conditions of Theorem 3.2 for any $0<L<R$,

$\displaystyle\int_{0}^{L}r^{D-k-1}R_{k,L}(r)dr$
$\displaystyle=\int_{0}^{R}r^{D-k-1}R_{k}(r)dr.$

###### Proof.

Observe that

$\displaystyle\int_{0}^{L}r^{D-k-1}R_{k,L}(r)dr=\int_{0}^{R}r^{D-k-1}R_{k}(r)dr-\int_{L}^{R}r^{D-k-1}\mathbb{E}C_{k}(F(r))dr$
$\displaystyle+\int_{0}^{\infty}r^{D-k-1}\mathbb{E}\sum_{i=1}^{N}\mathbf{1}_{(Lr_{i},Rr_{i}]}(r)C_{k}(F_{i}(r))dr=:I_{1}-I_{2}+I_{3}\,,$

provided that all three integrals on the right side converge absolutely. For
the first integral $I_{1}$ this is included in the proof of Theorem 3.2 for
$L=R$, and for $I_{2}$ it follows from the estimate (4.2). Replacing in
$I_{3}$ the curvature measure $C_{k}$ by $|C_{k}|$ and interchanging
expectation and integration we obtain

$\displaystyle\int_{0}^{\infty}r^{D-k-1}$
$\displaystyle\mathbb{E}\sum_{i=1}^{N}\mathbf{1}_{(Lr_{i},Rr_{i}]}(r)|C_{k}(F_{i}(r))|dr$
$\displaystyle=\mathbb{E}\sum_{i=1}^{N}\int_{0}^{\infty}r^{D-k-1}\mathbf{1}_{(Lr_{i},Rr_{i}]}(r)|C_{k}(F_{i}(r))|dr$
$\displaystyle=\mathbb{E}\sum_{i=1}^{N}\int_{0}^{\infty}r^{D-k-1}\mathbf{1}_{(L,R]}(r/r_{i})r_{i}^{k}|C_{k}(F(r/r_{i}))|dr$
$\displaystyle=\mathbb{E}\sum_{i=1}^{N}r_{i}^{D}\int_{L}^{R}\bar{r}^{D-k-1}|C_{k}(F^{(1)}(\bar{r}))|d\bar{r}$
$\displaystyle=\mathbb{E}\sum_{i=1}^{N}r_{i}^{D}\mathbb{E}\int_{L}^{R}\bar{r}^{D-k-1}|C_{k}(F^{(1)}(\bar{r}))|d\bar{r}$
$\displaystyle=\int_{L}^{R}\bar{r}^{D-k-1}\mathbb{E}|C_{k}(F(\bar{r}))|d\bar{r},$

where we have substituted $\bar{r}=r/r_{i}$ in the second step and used that
$F_{i}=f_{i}(F^{(1)})$. For the third step notice that $F^{(1)}$ is
independent of the first step of the construction. Hence the expectation can
be written as a product of two expectations, where the first one
$\mathbb{E}\sum_{i=1}^{N}r_{i}^{D}$ equals 1, by the definition of $D$. In the
second one we interchanged again expectation and integration. The last
integral is finite because of (4.2). Hence, we have shown the existence of the
integral $I_{3}$. Now we can repeat the last transformations omitting the
absolute value signs in order to get $I_{3}=I_{2}$. Together with the above
decomposition this proves the assertion. ∎

##  5\. Sufficient conditions and examples

We discuss some examples to illustrate our main result and compare it with the
known results in the random recursive case. In order to simplify the
verification of the conditions (ii) and (iii) in Theorem 3.2, we discuss first
some simpler sufficient conditions, which may equally be used in the random
recursive case. Recall that $O_{\sigma}=f_{\sigma}(O)$ for any word
$\sigma\in\Sigma_{*}$. As a first step, the following observation clarifies
that not too many of the parallel sets $O_{\sigma}(r)$, $\sigma\in\Sigma(r)$
intersect. Recall also that we assume here a uniform lower bound $r_{\min}$
for the contraction ratios, cf. the first lines of Section 3. For a similar
estimate in the deterministic case see e.g. [Wi08, Lemma 5.3.1].

######  Lemma 5.1.

Let $F$ be a homogeneous random fractal or a random self-similar set
satisfying UOSC (3.3) for some open set $O$. Then there is a constant
$\Gamma>0$ such that $\mathbb{P}$-a.s. for all $r>0$ and all
$\sigma\in\Sigma(r)$,

$\displaystyle\sharp\\{\sigma^{\prime}\in\Sigma(r):O_{\sigma}(r)\cap
O_{\sigma^{\prime}}(r)\neq\emptyset\\}\leq\Gamma\,.$

###### Proof.

We provide a proof here for the homogeneous model. The one for the random
recursive model is literally the same, when $\mathbb{P}$ is replaced by the
corresponding measure, see, e.g., [Za11, eq. (9)]. Let $O$ be the open set in
UOSC. The condition implies that $\mathbb{P}$-a.s. for any $r>0$ the family
$\\{O_{\sigma}:\sigma\in\Sigma(r)\\}$ consists of pairwise disjoint sets. The
definitions of $R>\sqrt{2}|O|$ and $\Sigma(r)$ imply that for any
$\sigma\in\Sigma(r)$

(5.1)
$\displaystyle|O_{\sigma}|=r_{\sigma}|O|\leq\frac{1}{\sqrt{2}}Rr_{\sigma}\leq\frac{r}{\sqrt{2}}.$

Now fix $r>0$ and let $\sigma,\sigma^{\prime}\in\Sigma(r)$ such that
$O_{\sigma}(r)\cap O_{\sigma^{\prime}}(r)\neq\emptyset$. Then, by (5.1),

$\displaystyle O_{\sigma^{\prime}}\subset
O_{\sigma}(2r+|O_{\sigma^{\prime}}|)\subset
B\left(f_{\sigma}(x),(2+\sqrt{2})r\right)\subset
B\left(f_{\sigma}(x),4r\right),$

where $x$ is an arbitrary point in $O$, and thus $f_{\sigma}(x)\in
O_{\sigma}$. Here $B(y,s)$ denotes the closed ball with center $y$ and radius
$s$. Recalling that the volume of a ball of radius $4r$ is
$\kappa_{d}(4r)^{d}$ (where $\kappa_{d}$ is the volume of the unit ball in
$\mathbb{R}^{d}$) and that the volume of each of the (pairwise disjoint) sets
$O_{\sigma^{\prime}}$, $\sigma^{\prime}\in\Sigma(r)$ is bounded from below by

$\displaystyle
C_{d}(O_{\sigma^{\prime}})=r_{\sigma^{\prime}}^{d}C_{d}(O)>C_{d}(O)R^{-d}r_{\min}^{d}r^{d},$

we conclude that not too many of the sets $O_{\sigma^{\prime}}$ can be
contained in the ball $B\left(f_{\sigma}(x),4r\right)$. Hence we obtain that
$\mathbb{P}$-almost surely

$\displaystyle\sharp\left\\{\sigma^{\prime}\in\Sigma(r):O_{\sigma}(r)\cap
O_{\sigma^{\prime}}(r)\neq\emptyset\right\\}$
$\displaystyle\leq\sharp\left\\{\sigma^{\prime}\in\Sigma(r):O_{\sigma^{\prime}}\subseteq
B(f_{\sigma}(x),4r)\right\\}$
$\displaystyle\leq\frac{\kappa_{d}4^{d}R^{d}}{r_{\min}^{d}C_{d}(O)}=:\Gamma,$

where the constant $\Gamma$ is independent of $r>0$ and $\sigma$. ∎

Observe that, by UOSC, a.s. $F\subset\overline{O}$, which implies

(5.2) $\displaystyle F(r)\subset O(r)\quad\mathbb{P}\text{-a.s. for any }r>0$

and hence, $F_{\sigma}(r)\subset O_{\sigma}(r)$ for any $\sigma\in\Sigma_{*}$.
Therefore, the assertion of Lemma 5.1 does equally hold with the sets
$O_{\sigma}(r)\cap O_{\sigma^{\prime}}(r)$ replaced by $F_{\sigma}(r)\cap
F_{\sigma^{\prime}}(r)$.

Now we formulate the announced conditions that imply (ii) and (iii) in Theorem
3.2. They are almost sure bounds in contrast to the bounds on expectations in
(ii) and (iii). In the deterministic case, a condition very similar to the one
in (b) is known to be equivalent to the curvature bounds corresponding to (ii)
and (iii), cf. [Wi11, Thm. 4.7].

######  Proposition 5.2.

Let $k\in\\{0,1,\ldots,d\\}$ and let $F$ be a homogeneous random fractal or a
random self-similar set satisfying USOSC and the regularity condition (i) in
Theorem 3.2. Suppose that one of the following equivalent conditions (a)–(d)
holds:

(a) there is a constant $c>0$ such that $\mathbb{P}$-a.s. for almost all $r>0$
and all $\sigma\in\Sigma(r)$,

$\displaystyle
C_{k}^{\operatorname{var}}\bigg{(}F(r),F_{\sigma}(r)\cap\bigcup_{\tau\in\Sigma(r)\setminus\\{\sigma\\}}F_{\tau}(r)\bigg{)}\leq
cr^{k};$

(b) there is $c^{\prime}>0$ such that $\mathbb{P}$-a.s. for a.a. $r>0$ and all
$\sigma,\sigma^{\prime}\in\Sigma(r)$ with $\sigma\neq\sigma^{\prime}$,

$\displaystyle C_{k}^{\operatorname{var}}(F(r),F_{\sigma}(r)\cap
F_{\sigma^{\prime}}(r))\leq c^{\prime}r^{k};$

(c) there is $c^{\prime\prime}>0$ such that $\mathbb{P}$-a.s. for a.a. $r>0$
and all $\sigma\in\Sigma(r)$,

$\displaystyle C_{k}^{\operatorname{var}}(F(r),F_{\sigma}(r))\leq
c^{\prime\prime}r^{k};$

(d) there is $c^{\prime\prime\prime}>0$ such that $\mathbb{P}$-a.s. for a.a.
$r>0$ and all $\sigma,\sigma^{\prime}\in\Sigma(r)$ with
$\sigma\neq\sigma^{\prime}$,

$\displaystyle C_{k}^{\operatorname{var}}(F(r),O_{\sigma}(r)\cap
O_{\sigma^{\prime}}(r))\leq c^{\prime\prime\prime}r^{k}.$

Then the conditions (ii) and (iii) in Theorem 3.2 are satisfied.

###### Proof.

Again we provide a proof for the homogeneous model, the one for the recursive
case being similar. First we show the equivalence of the four conditions
(a)–(d). The implications (a) $\Rightarrow$ (b), (c) $\Rightarrow$ (b) and (d)
$\Rightarrow$ (b) are obvious from corresponding set inclusions.

(b) $\Rightarrow$ (a): Suppose (b) holds. Then $\mathbb{P}$-a.s. for a.a.
$r>0$ and any $\sigma\in\Sigma(r)$,

$\displaystyle C_{k}^{\operatorname{var}}\bigg{(}\,$ $\displaystyle
F(r),F_{\sigma}(r)\cap\bigcup_{\tau\in\Sigma(r)\setminus\\{\sigma\\}}F_{\tau}(r)\bigg{)}=C_{k}^{\operatorname{var}}\bigg{(}F(r),\bigcup_{\tau\in\Sigma(r)\setminus\\{\sigma\\}}F_{\sigma}(r)\cap
F_{\tau}(r)\bigg{)}$
$\displaystyle\leq\sum_{\tau\in\Sigma(r)\setminus\\{\sigma\\}}C_{k}^{\operatorname{var}}\big{(}F(r),F_{\sigma}(r)\cap
F_{\tau}(r)\big{)}\leq\Gamma\cdot c^{\prime}r^{k},$

since, by Lemma 5.1, the number of nonzero summands in the last sum is bounded
by some constant $\Gamma$ independent of $r$ or $\sigma$. Moreover, by (b),
each of these summands is bounded by $c^{\prime}r^{k}$. Hence, (b) implies (a)
(with constant $c=\Gamma c^{\prime}$), and therefore both conditions are
equivalent.

(a) $\Rightarrow$ (c): We have $\mathbb{P}$-a.s., for a.a. $r>0$ and
$\sigma\in\Sigma(r)$,

$\displaystyle C_{k}^{\operatorname{var}}$
$\displaystyle\left(F(r),F_{\sigma}(r)\right)\leq
C_{k}^{\operatorname{var}}\left(F(r),F_{\sigma}(r)\cap
U_{r,\sigma}\right)+C_{k}^{\operatorname{var}}\left(F(r),F_{\sigma}(r)\setminus
U_{r,\sigma}\right),$

where
$U_{r,\sigma}:=\bigcup_{\tau\in\Sigma(r)\setminus\\{\sigma\\}}F_{\tau}(r)$. By
(a), the first summand on the right is bounded by $cr^{k}$. For the second
summand we infer that

$\displaystyle C_{k}^{\operatorname{var}}\left(F(r),F_{\sigma}(r)\setminus
U_{r,\sigma}\right)\leq
C_{k}^{\operatorname{var}}\left(F(r),\mathbb{R}^{d}\setminus
U_{r,\sigma}\right)=C_{k}^{\operatorname{var}}\left(F_{\sigma}(r),\mathbb{R}^{d}\setminus
U_{r,\sigma}\right),$

by the local definiteness, and therefore, by Lemma 2.1,

$\displaystyle C_{k}^{\operatorname{var}}\left(F(r),F_{\sigma}(r)\setminus
U_{r,\sigma}\right)\leq
C_{k}^{\operatorname{var}}\left(F_{\sigma}(r)\right)\leq c_{k}(R)r^{k},$

since, by definition of $\Sigma(r)$, $r\geq
Rr_{\sigma}>\sqrt{2}|O|r_{\sigma}\geq\sqrt{2}|F_{\sigma}|$. Hence (c) holds
(with $c^{\prime\prime}=c+c_{k}(R)$), showing the equivalence of (a) and (c).

(c) $\Rightarrow$ (d): We have $\mathbb{P}$-a.s., for a.a. $r>0$ and all
$\sigma,\sigma^{\prime}\in\Sigma(r)$ with $\sigma\neq\sigma^{\prime}$, (with
$U_{r,\sigma}$ defined as above and noting that $F(r)=F_{\sigma}(r)\cup
U_{r,\sigma}$)

$\displaystyle C_{k}^{\operatorname{var}}$
$\displaystyle\left(F(r),O_{\sigma}(r)\cap
O_{\sigma^{\prime}}(r)\right)=C_{k}^{\operatorname{var}}\left(F(r),O_{\sigma}(r)\cap
O_{\sigma^{\prime}}(r)\cap\left(F_{\sigma}(r)\cup U_{r,\sigma}\right)\right)$
$\displaystyle\leq C_{k}^{\operatorname{var}}\left(F(r),O_{\sigma}(r)\cap
O_{\sigma^{\prime}}(r)\cap
F_{\sigma}(r)\right)+C_{k}^{\operatorname{var}}\left(F(r),O_{\sigma}(r)\cap
O_{\sigma^{\prime}}(r)\cap U_{r,\sigma}\right)$ $\displaystyle\leq
C_{k}^{\operatorname{var}}\left(F(r),F_{\sigma}(r)\right)+C_{k}^{\operatorname{var}}\left(F(r),O_{\sigma}(r)\cap
U_{r,\sigma}\right)$

Due to (c), the first summand is bounded by $c^{\prime\prime}r^{k}$, while for
the second summand we get

$\displaystyle C_{k}^{\operatorname{var}}\left(F(r),O_{\sigma}(r)\cap
U_{\sigma,r}\right)$ $\displaystyle\leq
C_{k}^{\operatorname{var}}\left(F(r),\bigcup_{\tau\in\Sigma(r)\setminus\\{\sigma\\},O_{\tau}(r)\cap
O_{\sigma}(r)\neq\emptyset}F_{\tau}(r)\right)$
$\displaystyle\leq\sum_{\tau\in\Sigma(r)\setminus\\{\sigma\\},O_{\tau}(r)\cap
O_{\sigma}(r)\neq\emptyset}C_{k}^{\operatorname{var}}\left(F(r),F_{\tau}(r)\right).$

Now, again by (c), each summand is bounded by $c^{\prime\prime}r^{k}$ and, by
Lemma 5.1, the number of summands is bounded by some constant $\Gamma$. Hence
(d) holds (with $c^{\prime\prime\prime}=(1+\Gamma)c^{\prime\prime}$).

It suffices now to show that condition (a) implies (ii) and (iii) in Theorem
3.2. Applying condition (a) to the expectation in condition (ii), we conclude
that for any $r_{0}\in(0,R)$, this expectation is bounded from above by

$\displaystyle\mathbb{E}\operatorname{ess\,sup}_{r_{0}\leq r\leq
R}\limits\,\max_{\sigma\in\Sigma_{b}(r)}cr^{k}\leq cR^{k}<\infty.$

Similarly, the conditional expectation in condition (iii) is bounded from
above by

$\displaystyle\mathbb{E}\bigg{[}\max_{\sigma\in\Sigma_{b}(r)}\,r^{-k}\,cr^{k}\bigg{|}\sharp\Sigma_{b}(r)\bigg{]}\leq
c\Gamma$

for a.a. $r\in(0,R)$. Hence conditions (ii) and (iii) are satisfied as
claimed. ∎

Figure 1.  Illustration of the first construction step of the two IFS $G$ and
$H$ in Example 5.3 used to generate the homogeneous random Sierpiński gasket.

######  Example 5.3.

(Homogeneous random Sierpiński gasket) Let $G=(g_{1},g_{2},g_{3})$ be the IFS
of the standard Sierpiński gasket in $\mathbb{R}^{2}$, i.e.
$g_{i}(x)=\frac{1}{2}x+t_{i}$, $x\in\mathbb{R}^{2}$, $i=1,2,3$, where
$t_{1}=(0,0)$, $t_{2}=(\frac{1}{2},0)$ and
$t_{3}=(\frac{1}{4},\frac{\sqrt{3}}{4})$, and let $H=(h_{1},\ldots,h_{6})$ be
the IFS of the modified Sierpiński gasket given by the six mappings
$h_{i}(x)=\frac{1}{3}x+s_{i}$, $x\in\mathbb{R}^{2}$, $i=1,\ldots,6$, where
$s_{1}=(0,0)$, $s_{2}=(\frac{1}{3},0)$, $s_{3}=(\frac{2}{3},0)$,
$s_{4}=(\frac{1}{6},\frac{\sqrt{3}}{6})$,
$s_{5}=(\frac{1}{2},\frac{\sqrt{3}}{6})$ and
$s_{6}=(\frac{1}{3},\frac{\sqrt{3}}{3})$, see also Figure 1. Then, for any
$p\in[0,1]$, we consider the distribution $\mathbb{P}_{0}$ on the space
$\Omega_{0}$ of the primary random IFS with

(5.3)
$\displaystyle\mathbb{P}_{0}(\\{G\\})=p,\qquad\mathbb{P}_{0}(\\{H\\})=1-p\,.$

Let $F$ be the corresponding homogeneous random fractal generated as in
Section 3. Keep in mind that $\mathbb{P}_{0}$ and $F$ depend on the parameter
$p$, although we suppress this dependence in the notation. The two
deterministic self-similar sets generated by $G$ and $H$ are contained in this
one-parameter family as marginal cases corresponding to $p=1$ and $p=0$,
respectively.

First note that $F$ satisfies UOSC for the open set $O=\operatorname{int}(T)$,
where
$T:=\operatorname{conv}\\{(0,0),(1,0),(\frac{1}{2},\frac{\sqrt{3}}{2})\\}$,
since both $G$ and $H$ satisfy OSC for $O$. Furthermore, $3<N\leq 6$ with
probability 1. We claim that the regularity condition (i) in Theorem 3.2 is
satisfied. In fact, $\mathbb{P}$-a.s. all parallel sets of $F$ are polyconvex.
(To see this, note that $\partial T\subset F\subset T$ and that for any
$r\geq|T|=|F|$, $F(r)=T(r)$. Hence $F(r)$ is convex for these $r$. Now let
$r>0$ be arbitrary and choose $n\in\mathbb{N}$ such that for all
$\sigma\in\Sigma_{n}$, $|F_{\sigma}|\leq r$. (This is possible, since in each
step sets are contracted at least by the factor $\frac{1}{2}$.) Then
$F_{\sigma}(r)=T_{\sigma}(r)$ for all $\sigma\in\Sigma_{n}$ and, since
$F(r)=\bigcup_{\sigma\in\Sigma_{n}}F_{\sigma}(r)$, we have found a
representation of $F(r)$ by a finite number of convex sets. Hence $F(r)$ is
polyconvex for each $r>0$.)

In order to verify conditions (ii) and (iii), it suffices to check the
assumptions of Proposition 5.2. First we need to specify the constant $R$. It
is convenient to choose $R=\frac{3}{2}$. Since $|O|=1$, it clearly satisfies
$R>\sqrt{2}|O|$. Let $r>0$ and recall the definition of the family
$\Sigma(r)$. In the present example, the sets $F_{\sigma}$,
$\sigma\in\Sigma(r)$, are all congruent copies of each other. In particular,
they have the same diameter $r_{\sigma}$ and their parallel sets
$F_{\sigma}(r)=T_{\sigma}(r)$ are convex, since $r\geq
Rr_{\sigma}>\sqrt{2}|F_{\sigma}|$.

Recall that $F(r)=\bigcup_{\tau\in\Sigma(r)}F_{\tau}(r)$ is a representation
of $F(r)$ by convex sets. By Lemma 5.1, there is a constant $\Gamma$ such that
locally within a fixed set $F_{\sigma}(r)$, $F(r)$ can be represented by at
most $\Gamma$ of these sets. Let
$\Sigma_{\sigma}(r):=\\{\tau\in\Sigma(r):F_{\tau}(r)\cap
F_{\sigma}(r)\neq\emptyset\\}$. By the above considerations,
$\sharp\Sigma_{\sigma}(r)\leq\Gamma$ and, since curvature measures are locally
defined, we get

$\displaystyle C_{k}^{\operatorname{var}}(F(r),F_{\sigma}(r)\cap
F_{\sigma^{\prime}}(r))$
$\displaystyle=C_{k}^{\operatorname{var}}\bigg{(}\bigcup_{\tau\in\Sigma_{\sigma}(r)}F_{\tau}(r),F_{\sigma}(r)\cap
F_{\sigma^{\prime}}(r)\bigg{)}$ $\displaystyle\leq
C_{k}^{\operatorname{var}}\bigg{(}\bigcup_{\tau\in\Sigma_{\sigma}(r)}F_{\tau}(r)\bigg{)}$
$\displaystyle\leq
2^{\Gamma}\max_{\tau\in\Sigma_{\sigma}(r)}C_{k}(F_{\tau}(r))$
$\displaystyle\leq
2^{\Gamma}\max_{\tau\in\Sigma_{\sigma}(r)}r_{\tau}^{k}C_{k}(F^{(|\tau|)}(r^{-1}_{\tau}r))$
$\displaystyle\leq 2^{\Gamma}R^{-k}r^{k}C_{k}(B(0,1+6R))=:cr^{k},$

where in the third step we used [Wi08, Lemma 3.1.4] and in the last line the
monotonicity of total curvatures for convex sets together with the fact that
$|F(r_{\tau}^{-1}r)|\leq|F|+2r_{\tau}^{-1}\leq 1+2Rr_{\min}^{-1}\leq 1+6R$.
Hence, by Proposition 5.2, conditions (ii) and (iii) of Theorem 3.2 are
satisfied and we can apply this theorem.

Observe that the scaling exponent $D$ is given by the equation

(5.4) $\displaystyle 3p\cdot 2^{-D}+6(1-p)\cdot 3^{-D}=1,$

and that the distribution of the logarithmic contraction ratios is

$\displaystyle\mu=\frac{3p}{2^{D}}\mathbf{1}_{(\cdot)}(\ln(2))+\frac{6(1-p)}{3^{D}}\mathbf{1}_{(\cdot)}(\ln(3)).$

Hence $\mu$ is non-lattice for $p\in(0,1)$ (and lattice for $p\in\\{0,1\\}$).
Its mean value is given by

(5.5) $\displaystyle\eta=\frac{3p}{2^{D}}\ln(2)+\frac{6(1-p)}{3^{D}}\ln(3).$

In order to compute the limits of the expected rescaled total curvatures, as
provided by Theorem 3.2, it remains, to determine the functions $R_{k,L}$ for
$k=0,1,2$.

For the computations it is convenient to set $L:=\frac{\sqrt{3}}{6}$ (which is
the inradius of an equilateral triangle with sidelength 1). We split the
interval $(0,L)$ into three pieces. For $r\in[L/2,L)$, all indicators in
$R_{k,L}$ are zero almost surely, since $r_{i}\leq\frac{1}{2}$ a.s. Hence
$R_{k,L}(r)=\mathbb{E}C_{k}(F(r))=C_{k}(T(r))$ in this case, since $r$ is
large enough such that $F(r)$ has no holes. For
$r\in[\frac{L}{3},\frac{L}{2}]$, there are two possible situations. If we
condition on the event that in the first step of the construction the IFS $G$
is chosen, then there will be a hole in $F(r)$. Moreover, the indicators in
$R_{k,L}$ are 1, $N=3$, $r_{i}=\frac{1}{2}$ and the sets
$F_{i}(r)=(g_{i}T)(r)$ are convex. Since $F(r)=\bigcup_{i=1}^{3}F_{i}(r)$ in
this case, the inclusion-exclusion principle implies

$\displaystyle\mathbb{E}\big{[}\xi_{k}^{L}(r)\big{|}(f_{1},\ldots,f_{N})=G\big{]}$
$\displaystyle=\mathbb{E}\bigg{[}C_{k}(F(r))-\sum_{i=1}^{3}C_{k}(F_{i}(r))\bigg{|}(f_{1},\ldots,f_{N})=G\bigg{]}$
$\displaystyle=-3C_{k}((g_{1}T)(r)\cap(g_{2}T)(r)).$

Otherwise, i.e. if $H$ is chosen, we have
$\mathbb{E}\big{[}\xi_{k}^{L}(r)\big{|}(f_{1},\ldots,f_{N})=H\big{]}=C_{k}(T(r))$.
This yields

$\displaystyle R_{k,L}(r)=(1-p)C_{k}(T(r))-p\cdot
3C_{k}((g_{1}T)(r)\cap(g_{2}T)(r)),\qquad r\in[\frac{L}{3},\frac{L}{2}].$

Finally, for $r\in(0,L/3)$, all indicators in $R_{k,L}$ are 1 a.s. and
recalling that $F=\bigcup_{i=1}^{N}F_{i}$, the inclusion-exclusion principle
implies that

$\displaystyle R_{k,L}(r)=\mathbb{E}\sum_{I\subset[N],\sharp I\geq
2}(-1)^{\sharp I-1}C_{k}\bigg{(}\bigcap_{i\in I}F_{i}(r)\bigg{)},\qquad
r\in(0,\frac{L}{3}),$

where the summation is over all subsets $I$ of $[N]:=\\{1,\ldots,N\\}$ with at
least two elements. Conditioning on either $G$ or $H$ being chosen in the
first step of the construction and taking into account the symmetries of the
resulting intersections, it is easily seen that

$\displaystyle-R_{k,L}(r)=p$ $\displaystyle\cdot 3\
C_{k}((g_{1}T)(r)\cap(g_{2}T)(r)))+(1-p)\cdot$
$\displaystyle\left(9C_{k}((h_{1}T)(r)\cap(h_{2}T)(r))-C_{k}((h_{2}T)(r)\cap(h_{4}T)(r)\cap(h_{5}T)(r))\right).$

for any $r\in(0,L/3)$. Here we have used that under the condition
$(f_{1},\ldots,f_{N})=G$, $F_{1}(r)\cap F_{2}(r)=(g_{1}T)(r)\cap(g_{2}T)(r)$
holds for any $r\in(0,L/3)$, and analogous relations under the condition
$(f_{1},\ldots,f_{N})=H$.

Note that all the sets occurring above in the expressions for $R_{k,L}$ are
nonempty and convex. For $k=0$, this gives

$\displaystyle R_{0,L}(r)$ $\displaystyle=\begin{cases}5p-8,&r\in(0,L/3),\\\
1-4p,&r\in[L/3,L/2),\\\ 1,&r\in[L/2,L).\end{cases}$

Hence, for $p\in(0,1)$, Theorem 3.2 (I) yields

$\displaystyle C_{0,F}^{\operatorname{frac}}=\lim_{\varepsilon\searrow
0}\varepsilon^{D}\mathbb{E}C_{0}(F(\varepsilon))=\frac{L^{D}}{D\cdot\eta}\left(1-(1-p)3^{2-D}-p2^{2-D}\right),$

where $L=\frac{\sqrt{3}}{6}$, $D$ is given by (5.4) and $\eta$ by (5.5). For
$p\in\\{0,1\\}$, part (III) of Theorem 3.2 yields the same value for
$\overline{C}_{0,F}^{\operatorname{frac}}$ (while the limit
$C_{0,F}^{\operatorname{frac}}$ does not exist in these two cases). For $p=1$,
this recovers the value $-\frac{L^{D}}{3\ln 3}$ obtained in [Wi08, Ex. 2.4.1].

For $k=1$, recall that $C_{1}$ is half the boundary length. Therefore, we have
for any $r>0$, $C_{1}(T(r))=\frac{3}{2}+\pi r$,

$\displaystyle
C_{1}((g_{1}T)(r)\cap(g_{2}T)(r))=C_{1}((h_{1}T)(r)\cap(h_{2}T)(r))=(\frac{2}{3}\pi+\sqrt{3})r$

and $C_{k}((h_{2}T)(r)\cap(h_{4}T)(r)\cap(h_{5}T)(r))=C_{1}(B(0,r))=\pi r$,
which yields

$\displaystyle R_{1,L}(r)$
$\displaystyle=\begin{cases}\left(3p(\pi+2\sqrt{3})-(9\sqrt{3}+5\pi)\right)\cdot
r=:c_{p}\cdot r,&r\in(0,L/3),\\\
\frac{3}{2}(1-p)+\underbrace{(\pi-3p(\pi+\sqrt{3})}_{=:\tilde{c}_{p}}\cdot
r,&r\in[L/3,L/2),\\\ \frac{3}{2}+\pi\cdot r,&r\in[L/2,L).\end{cases}$

Plugging this into the formula (I) in Theorem 3.2, we obtain

$\displaystyle C_{1,F}^{\operatorname{frac}}$
$\displaystyle=\lim_{\varepsilon\searrow
0}\varepsilon^{D-1}\mathbb{E}C_{1}(F(\varepsilon))=\frac{1}{\eta}\int_{0}^{L}r^{D-2}R_{1,L}(r)dr$
$\displaystyle=\frac{L^{D}}{D\eta}\left(3^{-D}(c_{p}-\tilde{c}_{p})+2^{-D}(\tilde{c}_{p}-\pi)+\pi\right)$
$\displaystyle\quad+\frac{3L^{D-1}}{2(D-1)\eta}\left(1-(1-p)3^{1-D}-p2^{1-D}\right).$

For $k=2$, we will derive below after Example 5.4 that
$C_{2,F}^{\operatorname{frac}}=\frac{2}{2-D}C_{1,F}^{\operatorname{frac}}$.

######  Example 5.4.

For $p\in(0,1)$, let $K=K_{p}$ be the random self-similar set generated by the
same IFS-distribution $\mathbb{P}_{0}=\mathbb{P}_{0,p}$ (given by (5.3)) as
that for the homogeneous random fractal $F=F_{p}$ in Example 5.3. (Note that
for $p\in\\{0,1\\}$ we get the same deterministic sets as above in Example
5.3.) By Remark 3.5, the mean fractal curvatures
$C_{k,K}^{\operatorname{frac}}$ of $K$ are given by the same formulas as those
of $F$. One just has to replace in the integrand the function $R_{k,L}$ for
$F$ by the corresponding one for $K$. It is not difficult to see that for all
$k$ the functions $R_{k,L}$ for $K$ and $F$ coincide. Indeed, going through
all the considerations in Example 5.3, it is clear that they apply equally to
$K$. Hence, by [Za11, Theorem 2.3.8], we get the same values for the mean
fractal curvatures of $K$ as for $F$, i.e.

$\displaystyle C_{k,K}^{\operatorname{frac}}=C_{k,F}^{\operatorname{frac}}$

for $k=0,1,2$ and any $p\in(0,1)$. Moreover, by the same theorem, the almost
sure limits $C_{k}(K):=\operatorname{ess\,lim}_{\varepsilon\searrow
0}\varepsilon^{D-k}C_{k}(K(\varepsilon))$ exist and coincide with
$C_{k,K}^{\operatorname{frac}}$, for any $p\in(0,1)$.

Recall from [RW13, Theorem 2.4] that, for any bounded set
$A\subset\mathbb{R}^{d}$ and any constants $s\in[0,d]$ and $M\in\mathbb{R}$,
the limit $\lim_{r\searrow 0}\frac{C_{d}(A(r))}{r^{d-s}}$ exists and equals
$M$ if and only if the limit $\lim_{r\searrow
0}\frac{2C_{d-1}(A(r))}{(d-s)r^{d-1-s}}$ exists and equals $M$. Therefore, for
any random self-similar set $K$ satisfying the assumptions of [Za11, Theorem
2.3.8], the (almost sure) existence of the limits $C_{k}(K)$ for $k=d,d-1$
implies that almost surely

$\displaystyle C_{d-1}(K)=\frac{d-D}{2}\lim_{r\searrow
0}\frac{2C_{d-1}(K(r))}{(d-D)r^{d-1-D}}=\frac{d-D}{2}\lim_{r\searrow
0}\frac{C_{d}(K(r))}{r^{d-D}}=\frac{d-D}{2}C_{d}(K)$

and thus also in the mean
$C_{d-1,K}^{\operatorname{frac}}=\frac{d-D}{2}C_{d,K}^{\operatorname{frac}}.$
For the self-similar random set $K=K_{p}$ in Example 5.4 we obtain

$\displaystyle
C_{1,K}^{\operatorname{frac}}=\frac{2-D}{2}C_{2,K}^{\operatorname{frac}},$

for any $p\in(0,1)$. Note that the observed coincidence of the mean fractal
curvatures of $K$ with those of the homogeneous random fractal $F$ in Example
5.3 implies now the same relation for $F$, i.e., we get similarly

$\displaystyle
C_{1,F}^{\operatorname{frac}}=\frac{2-D}{2}C_{2,F}^{\operatorname{frac}},$

as claimed above. Note that in contrast the limit $\lim_{\varepsilon\searrow
0}\varepsilon^{D-k}C_{k}(F(\varepsilon))$ (the corresponding $D$-dimensional
Minkowski content of $F$) vanishes almost surely. Indeed, the a.s. Minkowski
(and Hausdorff) dimension $D_{H}$ of $F$ is known to be given by the equation
$\mathbb{E}\ln\bigg{(}\sum_{i=1}^{N}r_{i}^{D_{H}}\bigg{)}=0$ and thus strictly
smaller than $D$, see e.g. [Za11]. It is an interesting open question whether
the relation
$C_{d-1,F}^{\operatorname{frac}}=\frac{d-D}{2}C_{d,F}^{\operatorname{frac}}$
holds for any homogeneous random fractal $F$ with $D<d$.

##  6\. Appendix

Given a nonempty compact set $K\subset\mathbb{R}^{d}$, we denote by

$d_{K}:\,x\mapsto d(x,K):=\inf_{a\in K}|x-a|,\quad x\in\mathbb{R}^{d},$

the distance function to $K$. Note that $d_{K}$ is a $1$-Lipschitz function.
Denote by

$\Sigma_{K}(x):=\\{a\in K:\,|x-a|=d_{K}(x)\\}$

the set of all closest points of $K$ from $x$. By [Fu85, Lemma 4.2], the
Clarke subgradient $\partial d_{K}(x)$ of $d_{K}$ at $x$ equals

$\partial d_{K}(x)=\operatorname{conv}(x-\Sigma_{K}(x)).$

Recall that $x$ is called a _regular point_ of $d_{K}$ if $0\not\in\partial
d_{K}(x)$. The above identity thus implies

(6.1) $x\not\in K\text{ is a regular point of }d_{K}\text{ if and only if
}x\not\in\operatorname{conv}\Sigma_{K}(x).$

Moreover, Fu [Fu85] showed that if $r>0$ is a _regular value_ of $d_{K}$
(i.e., all points $x$ with $d_{K}(x)=r$ are regular), then
$\operatorname{reach}\widetilde{K(r)}>0$.

Given a nonempty set $A\subset\mathbb{R}^{d}$, we will use the notation
$A^{o}:=\\{y\in\mathbb{R}^{d}:\,y\cdot a\leq 0\text{ for all }a\in A\\}$ for
the _polar cone_ of $A$. Note that the polar cone of the polar cone agrees
with the generated convex cone:

$A^{oo}=\left\\{\sum_{i=1}^{n}t_{i}a_{i}:\,t_{i}\geq 0,\,a_{i}\in
A,\,i=1,\dots,n,\,n\in\mathbb{N}\right\\}.$

If $\operatorname{reach}(\widetilde{K(r)},x)>0$, then the tangent and normal
cones to $\widetilde{K(r)}$ at $x$ fulfill

(6.2) $\displaystyle\operatorname{Tan}(\widetilde{K(r)},x)$
$\displaystyle=(\Sigma_{K}(x)-x)^{o},$ (6.3)
$\displaystyle\operatorname{Nor}(\widetilde{K(r)},x)$
$\displaystyle=(\Sigma_{K}(x)-x)^{oo}.$

Indeed, it is easy to see that if $a\in\Sigma_{K}(x)$, then
$\operatorname{int}B(a,|x-a|)\cap\widetilde{K(r)}=\emptyset$ and hence
$a-x\in\operatorname{Nor}(\widetilde{K(r)},x)$, by [RZ19, Lemma 4.5].
Moreover, if $u\in\operatorname{int}(\Sigma_{K}(x)-x)^{o}$, then the distance
to $K$ increases (locally) in direction $u$ from $x$ implying that there is
some $\varepsilon>0$, such that the whole segment $[x,x+\varepsilon u]$ is
contained in $\widetilde{K(r)}$. Hence,
$u\in\operatorname{Tan}(\widetilde{K(r)},x)$.

The above observations imply the characterization (2.9) of regular pairs
$\operatorname{Reg}$ (see (2.8)): If $r>0$ is a regular value of $d_{K}$ then
$\operatorname{reach}\widetilde{K(r)}>0$ and
$\operatorname{Nor}(\widetilde{K(r)},x)\cap\rho(\operatorname{Nor}(\widetilde{K(r)},x))$
is trivial by (6.3), hence $(r,K)\in\operatorname{Reg}$. For the reverse
inclusion, if $(r,K)\in\operatorname{Reg}$ for some $r>0$, then
$\operatorname{reach}\widetilde{K(r)}>0$ and if $x$ is a point with
$d_{K}(x)=r$ then
$\operatorname{Nor}(\widetilde{K(r)},x)\cap\rho(\operatorname{Nor}(\widetilde{K(r)},x))$
is trivial by the definition of $\operatorname{Reg}$ and, hence, due to (6.3),
$(\Sigma_{K}(x)-x)^{oo}$ contains no line through the origin, which implies by
(6.1) that $x\not\in\operatorname{conv}\Sigma_{K}(x)$, hence, $x$ is a regular
point of $d_{K}$. Consequently, $r$ is a regular value of $d_{K}$.

In the rest of this section we provide a proof of Lemma 2.4. Due to (2.9), it
can be reformulated as follows.

######  Theorem 6.1.

Let $K\subset\mathbb{R}^{d}$ be nonempty and compact and let $r_{0}>0$ be a
regular value of $d_{K}$. Then, for any $k=0,1,\dots,d$, the weak convergence

$\lim_{r\to r_{0}}C_{k}(K(r),\cdot)=C_{k}(K(r_{0}),\cdot)$

takes place.

We start with several auxiliary results. Given $A\subset\mathbb{R}^{d}$
nonempty and $r>0$, we denote

$A_{<r}:=\\{y\in\mathbb{R}^{d}:\,d_{A}(y)<r\\}.$

######  Lemma 6.2.

Let $r>0$, $x\in\mathbb{R}^{d}$ and a compact set $\Sigma\subset\partial
B(x,r)$ be given. Then

$\operatorname{int}B(s,|s-x|)\subset\Sigma_{<r},\quad
s\in\operatorname{conv}\Sigma.$

###### Proof.

Take any points $s\in\operatorname{conv}\Sigma$ and
$y\in\operatorname{int}B(s,|s-x|)$ and assume, for the contrary, that
$d(y,\Sigma)\geq r$. Let $H$ be the hyperplane of symmetry of $y$ and $x$ and
$H_{+}$ the closed half space with boundary $H$ and containing $x$. Then

$\Sigma\subset\partial B(x,r)\cap\widetilde{B(y,r)}\subset H_{+}.$

On the other hand, $H$ separates $x$ and $s$, hence, $s\not\in H_{+}$, which
contradicts the assumption $s\in\operatorname{conv}\Sigma$. ∎

######  Lemma 6.3.

If $K\subset\mathbb{R}^{d}$ is nonempty and compact and $r>0$ then

$\operatorname{reach}\widetilde{K(r)}\geq\inf\\{J_{K}(x):\,x\in\partial
K(r)\\},$

where

$J_{K}:x\mapsto\operatorname{dist}(x,\operatorname{conv}\Sigma_{K}(x)),\quad
x\in\mathbb{R}^{d}\setminus K.$

###### Proof.

Denote $\eta:=\inf\\{J_{K}(x):\,x\in\partial K(r)\\}$. If $\eta=0$ there is
nothing to prove. Thus, assume that $\eta>0$. We will show that

(6.4)
$d_{\operatorname{Tan}(\widetilde{K(r)},x)}(y-x)\leq\frac{|y-x|^{2}}{2\eta},\quad
x,y\in\widetilde{K(r)}.$

This will imply the assertion, see [RZ19, Proposition 4.14].

If $x\in\operatorname{int}\widetilde{K(r)}$ then
$\operatorname{Tan}(\widetilde{K(r)},x)=\mathbb{R}^{d}$ and (6.4) obviously
holds. Hence, assume that $x\in\partial\widetilde{K(r)}$, which implies that
$d_{K}(x)=r$.

Denote $T:=(\Sigma_{A}(x)-x)^{o}$, $N:=T^{o}$, and recall that
$T=\operatorname{Tan}(\widetilde{K(r)},x)$ and
$N=\operatorname{Nor}(\widetilde{K(r)},x)$ (see (6.2), (6.3)). Denote further
$u:=p_{T}(y-x)$ and $v:=p_{N}(y-x)$ ($p_{T}$, $p_{N}$ denote the orthogonal
projection to $T$, $N$, respectively), and note that $u\cdot v=0$ since $T,N$
are dual convex cones. We will show that

$|(y-x)-u|\leq\frac{|y-x|^{2}}{2\eta}$

whenever $y\in\widetilde{K(r)}$, which will prove (6.4). If $v=0$ then $y-x=u$
and we are done. If $v\neq 0$ denote $\bar{v}:=\frac{v}{|v|}$ and note then,
using elementary planar geometry,

$|(y-x)-u|=(y-x)\cdot\bar{v}.$

Hence, we have to show that

(6.5) $(y-x)\cdot\bar{v}\leq\frac{|y-x|^{2}}{2\eta}.$

Let $s\in\operatorname{conv}\Sigma_{K}(x)$ be such that $s-x=t\bar{v}$ for
some $t>0$. Applying Lemma 6.3 we obtain

$\operatorname{int}B(x+\eta\bar{v},\eta)\subset\operatorname{int}B(s,|s-x|)\subset
K_{<r}=\mathbb{R}^{d}\setminus\widetilde{K(r)},$

which means that

$\eta^{2}\leq|y-x-\eta\bar{v}|^{2}=|y-x|^{2}+\eta^{2}-2\eta(y-x)\cdot\bar{v},$

and this proves (6.5). ∎

######  Lemma 6.4.

The function $x\mapsto J_{K}(x)$ is lower semicontinuous on $\mathbb{R}^{d}$,
i.e.,

$\liminf_{y\to x}J_{K}(y)\geq J_{K}(x).$

###### Proof.

First we show that the set-valued function $x\mapsto\Sigma_{K}(x)$ is upper
semicontinuous (w.r.t. Hausdorff metric), i.e., that $\limsup_{y\to
x}\Sigma_{K}(y)\subset\Sigma_{K}(x)$. (This was proved in [RSM09, Lemma 5.1].
We repeat the short argument here for the convenience of the reader.) Let
$x_{n}\to x$, $a_{n}\in\Sigma_{K}(x_{n})$, $a_{n}\to a$. We will show that
$a\in\Sigma_{K}(x)$. If not, there would be another point $b\in K$ with
$|b-x|<|a-x|$. Let $n$ be sufficiently large that

$\max\\{|a_{n}-a|,|x_{n}-x|\\}<\varepsilon:=\frac{1}{3}(|a-x|-|b-x|).$

Then, by the triangle inequality,

$\displaystyle|x_{n}-b|$
$\displaystyle\leq|x_{n}-x|+|b-x|<\varepsilon+|a-x|-3\varepsilon$
$\displaystyle<|a-x|-|a-a_{n}|-\varepsilon\leq|a_{n}-x|-\varepsilon$
$\displaystyle<|a_{n}-x|-|x-x_{n}|\leq|a_{n}-x_{n}|,$

which means that $a_{n}$ is not the closest point of $K$ to $x_{n}$, a
contradiction.

Since the convex hull is a continuous operation on compact sets, also

$\limsup_{y\to
x}\operatorname{conv}(\Sigma_{K}(y)-y)\subset\operatorname{conv}(\Sigma_{K}(x)-x).$

This means that for any $\varepsilon>0$ there exists $\delta>0$ such that

$\operatorname{conv}(\Sigma_{K}(y)-y)\subset(\operatorname{conv}(\Sigma_{K}(x)-x))(\varepsilon),\quad|y-x|<\delta,$

which clearly implies that $J_{K}(y)\geq J_{K}(x)-\varepsilon$ whenever
$|y-x|<\delta$. ∎

###### Proof of Theorem 6.1.

For $k=d$ see Remark 2.5.

Assume now that $k\leq d-1$. Since the set of regular values of $d_{K}$ is
open, there exists an $\varepsilon>0$ such that for any
$r\in(r_{0}-\varepsilon,r_{0}+\varepsilon)$, $r$ is a regular value of
$d_{K}$, $\operatorname{reach}\widetilde{K(r)}>0$ and (2.3) holds. Thus, in
order to prove the statement, it is enough to show that

$\lim_{r\to
r_{0}}C_{k}(\widetilde{K(r)},\cdot)=C_{k}(\widetilde{K(r_{0})},\cdot).$

Since clearly $K(r)\to K(r_{0})$ in the Hausdorff distance as $r\to r_{0}$, it
will be enough to show that

(6.6) $\liminf_{r\to r_{0}}\operatorname{reach}\widetilde{K(r)}>0,$

and apply [Fe59, §5.9]. The function $J_{K}$ is positive on $\partial
K(r_{0})$ and, using Lemma 6.4, we obtain that for any $x\in\partial K(r_{0})$
there exists a $\delta(x)>0$ such that $J_{K}(y)>J_{K}(x)/2$ whenever $y\in
U(x,\delta_{x})$. By the compactness of $\partial K(r_{0})$, we easily find an
$\eta>0$ and an open set $U\subset\partial K(r_{0})$ such that $J_{K}>\eta$ on
$U$. Since $\partial K(r)\subset U$ for $r$ sufficiently close to $r_{0}$, we
conclude that

$\liminf_{r\to r_{0}}\,\inf\\{J_{K}(x):\,x\in\partial K(r)\\}\geq\eta,$

and the proof is completed by applying Lemma 6.3. ∎

At the end we show the measurability property used in the proof of Lemma 2.3.
By a _random signed measure_ we understand a mapping $\mu$ from a probability
space into the space of locally finite signed Borel measures such that
$\mu(B)$ is a random variable for any bounded Borel set $B$. Recall that the
space $\mathcal{K}$ of nonempty compact sets was provided with the Borel
$\sigma$-algebra determined by the Hausdorff distance.

######  Lemma 6.5.

Let $\mu=\mu^{\omega}$ be a random signed measure. Then the mapping

$\Phi:(\omega,K)\mapsto\mu^{\omega}(K),\quad(\omega,K)\in\Omega\times\mathcal{K}$

is jointly measurable.

###### Proof.

Note that the mapping

$\Psi:(\omega,f)\mapsto\int
f\,d\mu^{\omega},\quad(\omega,f)\in\Omega\times\mathcal{C}_{c}$

is jointly measurable ($\mathcal{C}_{c}$ is the space of continuous functions
with compact support with supremum metric). This follows from the fact that
$\omega\mapsto\int f\,d\mu^{\omega}$ is measurable and $f\mapsto\int
f\,d\mu^{\omega}$ is continuous, hence $\Psi$ is a Caratheodory function,
which is always jointly measurable, see [AB06, Lemma 4.51].

Further, given $K\in\mathcal{K}$ and $n\in\mathbb{N}$, consider the function

$f_{K,n}:x\mapsto(1-nd_{K}(x))^{+},\quad x\in\mathbb{R}^{d}.$

Since for any $K,K^{\prime}\in\mathcal{K}$ we have
$|f_{K,n}(x)-f_{K^{\prime},n}(x)|\leq nd_{H}(K,K^{\prime})$, the mapping
$\varphi_{n}:K\mapsto f_{K,n}$ is continuous from $\mathcal{K}$ to
$\mathcal{C}_{c}$. Thus, the mapping
$\tilde{\varphi}_{n}:(\omega,K)\mapsto(\omega,f_{K,n})$, as well as the
composition $\Phi_{n}:=\Psi\circ\tilde{\varphi}_{n}$, are measurable on the
product space $\Omega\times\mathcal{K}$. Since $\Phi_{n}(\omega,K)=\int
f_{K,n}\,d\mu^{\omega}\to\mu^{\omega}(K)=\Phi(\omega,K)$, $n\to\infty$, for
any $(\omega,K)$, the limit function $\Phi$ is jointly measurable as well. ∎

## References

  * [AB06] C.D. Aliprantis, K.C. Border: _Infinite Dimensional Analysis: A Hitchhiker’s Guide_. Third Ed., Springer, New York, 2006.
  * [As87] S. Asmussen: Applied Probabilities and Queues. J. Wiley, Chichester, (1987),
  * [BHS12] M. Barnsley, J. Hutchinson, Ö. Stenflo: V-variable fractals: dimension results. Forum Math. 24, no. 3, (2012), 445–470.
  * [Fa86] K. Falconer: Random fractals: Math. Proc. Camb. Phil. Soc. 100, (1986), 559–582.
  * [Fa95] K. Falconer: On the Minkowski measurability of fractals. Proc. Amer. Math. Soc. 123, (1995), 1115–1124.
  * [Fa97] K. Falconer: Techniques in Fractal Geometry. J. Wiley & Sons, Chichester, 1997.
  * [Fe59] H. Federer: Curvature measures. Trans. Amer. Math. Soc. 93, (1959), 418–491.
  * [Fe69] H. Federer: Geometric Measure Theory. Springer, 1969.
  * [Fe71] W. Feller: An Introduction to Probability Theory and Its Applications. Vol. II, 2nd Ed., J. Wiley, New York, 1971.
  * [Fu85] J.H.G. Fu: Tubular neighborhoods in Euclidean spaces. Duke Math. J., 52, (1985), 1025–1046.
  * [Ga00] D. Gatzouras: Lacunarity of self-similar and stochastically self-similar sets. Trans. Amer. Math. Soc. 352 (2000), no. 5, 1953–1983.
  * [Gr87] S. Graf: Statistically self-similar fractals. Probab. Theory Related Fields 74 (1987), 357–392.
  * [Ha92] B.M. Hambly: Brownian motion on a homogeneous random fractal. Probab. Theory Related Fields 94, (1992), 1–38.
  * [Hu81] J. Hutchinson: Fractals and self-similarity, Indiana Univ. Math. J. 30, (1981), 713–747.
  * [La88] S. Lalley: The packing and covering functions of self-similar fractals. Indiana Univ. Math. J. 37, (1988), 699–709.
  * [LRZ16] M. Lapidus, G. Radunović, D. Žubrinić: Fractal Zeta Functions and Fractal Drums. Higher-Dimensional Theory of Complex Dimensions, Springer International Publishing, Switzerland, 2016.
  * [Ma75] G. Matheron: Random Sets and Integral Geometry, J. Wiley, New York, 1975.
  * [MW86] R.D. Mauldin, S.C. Williams: Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, (1986), 325–346.
  * [RSS09] J. Rataj, V. Schmidt, E. Spodarev: On the expected surface area of the Wiener saussage. _Math. Nachr._ 282, (2009), 591–603.
  * [RSM09] J. Rataj, E. Spodarev, D. Meschenmoser: Approximations of the Wiener sausage and its curvature measures. _Ann. Appl. Probab._ 19, (2009), 1840–1859.
  * [RW10] J. Rataj, S. Winter: On volume and surface area of parallel sets. Indiana Univ. Math. J. 59, (2010), 1661–1686.
  * [RW13] J. Rataj and S. Winter: Characterization of Minkowski measurability in terms of surface area. J. Math. Anal. Appl. 400, (2013), 120–132.
  * [RZ03] J. Rataj and M. Zähle: Normal cycles of Lipschitz manifolds by approximation with parallel sets. Differential Geom. Appl. 19, (2003), 113–126.
  * [RZ05] J. Rataj and M. Zähle: General normal cycles and Lipschitz manifolds of bounded curvature. Ann. Global Anal. Geom. 27, (2005), 135–156.
  * [RZ19] J. Rataj, M. Zähle: Curvature Measures of Singular Sets, Springer Monographs in Math., 2019.
  * [RU11] M. Roy, M. Urbanski: Random graph directed Markov systems. Discrete Contin. Dyn. Syst. 30, (2011), 261–298.
  * [St76] L.L. Stachó: On the volume function of parallel sets. _Acta Sci. Math._ (Szeged) 38, (1976), 365–374.
  * [Tr17] S. Troscheit: On the dimension of attractors of random self-similar graph directed iterated function systems. J. Fractal Geom. 4, no. 3, (2017), 257–303.
  * [Tr21a] S. Troscheit: Exact Hausdorff and packing measures of random self-similar code trees with necks. Studia Mathematica, 257, no. 3, (2021), 253–285.
  * [Tr21b] S. Troscheit: On the Minkowski content of self-similar random homogeneous iterated function systems. Preprint 2021 (arXiv:2103.04664)
  * [Wi08] S. Winter: Curvature measures and fractals, Diss. Math. 453, (2008), 1–66.
  * [Wi11] S. Winter: Curvature bounds for neighborhoods of self-similar sets. Comment. Math. Univ. Carolin. 52, no. 2, (2011), 205–226.
  * [WZ13] S. Winter, M. Zähle: Fractal curvature measures of self-similar sets. Adv. Geom. 13, no. 2, (2013), 229–244.
  * [Za82] M. Zähle: Random processes of Hausdorff rectifiable closed sets. Math. Nachr. 108, (1982), 49–72.
  * [Za86] M. Zähle: Curvature measures and random sets II. Probab. Th. Rel. Fields, 71, (1986), 37–58.
  * [Za86b] M. Zähle: Integral and current representation of Federer’s curvature measures. Archiv Math. 46 (1986), 557–567.
  * [Za11] M. Zähle: Lipschitz-Killing curvatures of self-similar random fractals. Trans. Amer. Math. Soc. 363, (2011), 2663–2684.
  * [Za20] M. Zähle: The Mean Minkowski Content of Homogeneous Random Fractals. Mathematics. 8, (2020), 883.