$$\rho_{d-k}(y_1,\dots ,y_{d-k};\mu_a^{(d-k)})=c_{\nu ,a}e^{-a\lambda (Y)}\sum_{n=0}^\infty \frac{a^n}{n!}\int_Y\dots \int_Y$$$$\exp\left(\frac{\nu_{d-k}}{(d-k)!}\sum_{(x_1,\dots ,x_{d-k})\subset\{u_1,\dots ,u_n,y_1,\dots ,y_{d-k}\}}{\mathbb H}^k(\cap_{i=1}^{d-k}x_i)\right)\lambda^n ({\text{\rm d}}(u_1,\dots ,u_n)).$$

The normalizing constant can be expressed from $$1={\mathbb E} p(\eta_a)=c_{\nu ,a}e^{-a\lambda (Y)}\sum_{n=0}^\infty \frac{a^n}{n!}\int_Y\dots \int_Y$$$$\exp\left(\frac{\nu_{d-k}}{(d-k)!}\sum_{(x_1,\dots ,x_{d-k})\subset\{u_1,\dots ,u_n\}}{\mathbb H}^k(\cap_{i=1}^{d-k}x_i)\right)\lambda^n ({\text{\rm d}}(u_1,\dots ,u_n))$$ and the result follows. $\Box $

We obtain asymptotic results in a special model. Let $$\label{ymod}Y=[0,b]^d\times\{2b\}\times\left\{e_i,\;i=1,\dots ,d\right\},$$ where $e_i$ are canonical unit vectors. A facet is a set $$y=(z,\phi )=\{s\in D; \max_i|z_i-s_i|\leq b\},$$ $z$ denotes the centre and $D$ is hyperplane with normal orientation $\phi .$ That means facets have the same fixed size and shape and any non-parallel facets intersect. In the case of $d-k$ facets with different orientations we have bounds $$\label{boh}b^k\leq {\mathbb H}^k(\cap_{i=1}^{d-k}x_i)\leq (2b)^k.$$ In the intensity $\lambda $ (\[inti\]) we have $Q=\delta_{2b},$ the orientation distribution $V$ is uniform on $\{e_i,\;i=1,\dots ,d\}.$ Then for $u_i=(z_i,\phi_i),\,i=1,\dots ,n,$ it holds $$\label{rdf}
\lambda^n({\text{\rm d}}(u_1,\dots, u_n))=$$$$\chi(z_1){\text{\rm d}}z_1\dots \chi(z_n){\text{\rm d}}z_n{1\over d^n}{\sum\dots\sum}_{n_1+\dots +n_d=n}\frac{n!}{n_1!\dots n_d!}
\otimes_{i=1}^d\delta_{e_i}^{n_i}({\text{\rm d}}\phi_1,\dots,{\text{\rm d}}\phi_n).$$ Denote $T=\int_{[0,b]^d}\chi (z){\text{\rm d}}z.$

In the special model it can be shown that for any $l \in \{2, \ldots ,d \}$ if $\nu_{i} = 0, i \neq l$ and $\nu_{l} \geq 0$, then $p \not\in L^{1}(P_{\eta})$, thus the conditions applied to parameters are not only sufficient, but also neccesarry conditions for density existence.

