Rotational versions of the Crofton formula

Inspired by recent developments in stereology, rotational versions of the Crofton formula are derived. The first version involves rotation averages of Minkowski functionals. It is shown that for the special case where the Minkowski functional is surface area, the rotation average can be expressed in terms of hypergeometric functions. The second rotational version of the Crofton formula solves the .opposite. problem of finding functions with rotation averages equal to the Minkowski functionals. For the case of surface area, hypergeometric functions appear again. The second type of rotational Crofton formula has applications in local stereology. As a by-product, a formula involving mixed volumes is found.