Integral geometry of tensor valuations

Abstract

We prove a complete set of integral geometric formulas of Crofton type (involving integrations over affine Grassmannians) for the Minkowski tensors of convex bodies. Minkowski tensors are the natural tensor valued valuations generalizing the intrinsic volumes (or Minkowski functionals) of convex bodies. By Hadwiger's general integral geometric theorem, the Crofton formulas yield also kinematic formulas for Minkowski tensors. The explicit calculations of integrals over affine Grassmannians require several integral geometric and combinatorial identities. The latter are derived with the help of Zeilberger's algorithm.