On linear sections of lattice packings (Q1079818)

In d-dimensional Euclidean space \(E^ d\) let \({\mathfrak P}\) be a lattice packing of subsets of \(E^ d\), and let H be a k-dimensional linear subspace of \(E^ d\) \((0<k<d)\). Then, \({\mathfrak P}\) induces a packing in H consisting of all sets \(P\cap H\) with \(P\in {\mathfrak P}\). The relationship between the density of this packing in H and the density of \({\mathfrak P}\) is investigated. A result from the theory of uniform distribution of linear forms is used to prove an integral formula that enables one to valuate the density of the induced packing in H (under suitable assumptions on the sets of \({\mathfrak P}\) and the functionals used to define the densities). It is shown that this result leads to explicit formulas for the averages of the induced densities under the rotation of H. If the densities are taken with respect to the mean cross-sectional measures of convex bodies one obtains analogues of the integral geometric intersection formulas of Crofton.