The closed form of a rotational version of the famous Crofton formula is derived. In the case where the sectioned object is a compact $d$-dimensional $C^2$ manifold with boundary, the rotational average of intrinsic volumes measured on sections passing through a fixed point can be expressed as an integral over the boundary involving hypergeometric functions. In the more general case of a compact subset of $\mathbb{R}^d$ with positive reach, the rotational average also involves hypergeometric functions. For convex bodies, we show that the rotational average can be expressed as an integral with respect to a natural measure on supporting flats. It is an open question whether the rotational average of intrinsic volumes studied in the present paper can be expressed as a limit of polynomial rotation invariant valuations.

Keywords: Geometric measure theory, hypergeometric functions, integral geometry, intrinsic volume, stereology

2000 Mathematics Subject Classication: 60D05, 53C65, 52A22