Vertex numbers of weighted faces in Poisson hyperplane mosaics (Q603860)

In this paper the focus is on random mosaics generated by a stationary Poisson hyperplane process in \(\mathbb{R}^d\). The author considers \(L_j\)-weighted typical \(k\)-faces \(Z_{k,j}\), \(0\leq j\leq k\leq d\), where, for a \(d\)-dimensional polytope \(P\), \(L_j(P)\) denotes the \(j\)-dimensional Hausdorff measure of its \(j\)-skeleton. First, the expectation \(\mathbb{E}h(Z_{k,j})\) is computed, where \(h\) is a translation-invariant nonnegative measurable function on \(k\)-polytopes. Then the result is applied to the vertices of \(Z_{k,j}\). Sharp lower and upper bounds for their expected number are obtained.