of segment, surface processes, respectively. Rather than statistics we develop the facet process theory in arbitrary dimension and deal with limit behavior when the intensity of the reference Poisson process tends to infinity.

Recently in [@RefB] functionals of spatial point processes given by a density with respect to the Poisson process were investigated using the $L_2$ expansion from [@RefL] which is applied to the product of a functional and the density. Using a special class of functionals called $U$-statistics closed formulas for mixed moments of functionals are obtained. Similar formulas, but under a stronger assumption of a product form of the driving function of the functional, were derived in [@RefD] using the Georgii-Nguyen-Zessin formula. In processes with densities the key characteristics are the correlation functions [@RefG] of arbitrary order which are dual to kernel functions of the density as a function of the Poisson process.

We call facets some bounded subsets of hyperplanes with a given shape, size and orientation. Natural geometrical characteristics of the union of the facets, based on Hausdorff measure of the intersections of pairs, triplets, etc., of facets form $U$-statistics. Building a parametric density from exponential family, the limitations for the space of parameters have to 