  We present the multidimensional versions of the Pleijel and
Ambartzumian--Pleijel identities. We also obtain the generalization of both the
Blaschke--Petkantschin and Z\"ahle formulae considering the case when some
points are chosen inside the convex body and some on the boundary. Moreover, a
version of the Z\"ahle formula for the polytopes is derived.
