$k$-tuples is equal to $\lambda (Y)^k$ for each $k$ since we have a density $v$ in (\[inti\]). For the Hausdorff measure ${\mathbb H}^j$ in ${\mathbb R}^d$ of order $j$ and ${\mathbf x}\in{\mathbf N}$ (on $Y$) we put

$$\label{hrm}G_1({\mathbf x})=\sum_{y\in {\mathbf x}}{\mathbb H}^{d-1}(y),$$

$$G_2({\mathbf x})=\frac{1}{2}\sum_{(y_1,y_2)\in {\mathbf x}^2_{\neq } }{\mathbb H}^{d-2}(y_1\cap y_2),$$

$$G_{d-1}({\mathbf x})=\frac{1}{(d-1)!}\sum_{(y_1,\dots ,y_{d-1})\in {\mathbf x}^{d-1}_{\neq } }{\mathbb H}^1(\cap_{i=1}^{d-1}y_i),$$ $$G_{d}({\mathbf x})=\frac{1}{d!}\sum_{(y_1,\dots ,y_{d})\in {\mathbf x}^{d}_{\neq } }{\mathbb H}^0(\cap_{i=1}^{d}y_i).$$ We deal with $$\label{gseg}G({\mathbf x})=(G_1({\mathbf x}),\dots , G_d({\mathbf x})),$$ here $G_j$ is $U$-statistic of the $j-$th order, $j=1,\dots ,d.$ The facet process $\mu $ is defined by the density (\[den\]) $$p({\mathbf x})=c_\nu\exp(\nu\cdot G({\mathbf x}))$$ with respect to $\eta ,$ where $\nu = (\nu_1,\dots ,\nu_d)\in{\mathbb R}^d.$ For ${\mathbf x}=\{u_1,\dots ,u_n\}$ we have $$G_j({\mathbf x})\leq const.n^jb^{d-j},$$ which together with the assumption $\nu_j\leq 0,\,j=2,\dots ,d$ proves using (\[odhad\]) that $$p\in L_1(P_\eta )\cap L_2(P_\eta ).$$ In fact the assumption that the orientation distribution $V(d\phi )$ has a density $v$ can be weakened to the assumption that $V$ has at least $d$ atoms, which we will use later.

Specially for $d=3$ the facet process may serve as a model for platelike particles in materials microstructure of metals. Here $G_1$ yields the total area of all plates, $G_2$ is the 