It is shown how new integral-geometric formulae can be obtained from the
existing formulae of Crofton type. In particular, for classical Crofton
formulae in which the answer depends on the Riemannian volume, we obtain
generalizations in terms of the mixed Riemannian volume defined in the paper.
The method is based on the calculations in the ring of normal densities
constructed in the previous work of the authors.Comment: 16 pages, extending and improving on arXiv:1907.00633. Theorem 3 is
  ne