Some problems of geometric probability relative to hyperspheres in the Euclidean space \(E_ n\) (Q1203542)

Let \({\mathcal S}\) be a random hypersphere of diameter \(D\) in the Euclidean space \(E_ n\) with coordinates \(x_ i\) (\(i=1,\dots,n\)) and let \({\mathcal L}_ i\) be the lattices of hyperplanes whose equations are \(x_ i=ma_ i\) (\(m\in\mathbb{Z}\)); let us denote by \({\mathcal L}^ k\) (\(k=1,\dots,n\)) the lattices \({\mathcal L}^ k=\bigcup_{i=1}^ k {\mathcal L}_ i\). It is found the probability that \({\mathcal S}\) has \(r\) intersections with \({\mathcal L}^ k\), the expected value and the second moment of the number of hyperplanes of \({\mathcal L}^ k\) which intersect \({\mathcal S}\), the expected value of the sum of the \(m\)-th powers of the radius of the \((n-1)\)-dimensional hyperspheres intersections of \({\mathcal S}\) with \({\mathcal L}^ k\). Some of these results are generalized to the case in which \({\mathcal S}\) has not only a random position in the space, but also random diameter.