Integral identities for the boundary of a convex body

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