where $y_1,\dots ,y_m\in B\setminus{\mathbf x}$ are distinct, $$\begin{aligned}
Q_mG({\mathbf x})&=&D^m_{y_1,\dots,y_m}G({\mathbf x})\\
&&+\sum_{i_1,\dots,i_{m-1}\in\{1,\dots,m\}}D^{m-1}_{y_{i_1},\dots,y_{i_{m-1}}}G({\mathbf x})+\dots+
\sum_{1\leq i\leq m}D_{y_i}G({\mathbf x}).\end{aligned}$$

The intensity of the reference Poisson process depends on a specific model, see [@RefM] for interacting discs.

Facet processes with density
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Here we consider processes of interacting facets in ${\mathbb R}^d,\,d\in {\mathbb N},$ with their natural $U$-statistics. Let $$\label{bck}Y=B\times (0,b]\times {\mathbb S}^{d-1},$$ where $b>0$ is a size parameter, ${\mathbb S}^{d-1}$ is the hemisphere of axial orientations in ${\mathbb R}^d.$ For a point $y\in Y,\;y=(z,r,\phi )$ represents a facet which is a subset of a hyperplane $A=A(z,\phi)$ through point $z$ having normal orientation $\phi .$ Then $$\label{dist}y=\{s\in A;\, dist(s,z)\leq r\}$$ for a distance $dist$ in $A.$ The Poisson process $\eta $ on $Y$ has intensity measure $\lambda ,$ $$\label{inti}\lambda ({\text{\rm d}}(z, r, \phi ))=\chi (z){\text{\rm d}}zQ({\text{\rm d}}r)v(\phi )({\text{\rm d}}\phi ),$$ where $Q$ is the size distribution, a probability measure on $(0,b],$ $v$ is a probability density w.r.t. Lebesgue measure on ${\mathbb S}^{d-1},$ $\chi $ is a bounded intensity function of facet centres on $B.$

Let $k\in\{2,\dots ,d\}, $ we consider the intersection of any $k$ facets such that the corresponding hyperplanes are in general position, cf. [@RefS], p.133. The $\lambda^k$-measure of all such 