A New Integral Geometric Formula of the Blaschke‐Petkantschin Type

Recently, a new geometric measure decomposition has been derived by Zahle (1990), involving the r-fold product of the d-dimensional Hausdorff measure with itself. The application to moment measure estimation has been discussed in Jensen et al. (1990a) and Zahle (1990). The decomposition involves, in essence, the rotational invariant measure on r-dimensional linear subspaces in Rn and the r-fold product of the (d - n + r)-dimensional Hausdorff measure. In the present paper, we derive another decomposition of this type, involving linear subspaces containing a fixed lower-dimensional linear subspace. The formula has applications in geometric sampling.