Determination of the mean normal measure from isotropic means of flat sections

Let be the mean normal measure of a stationary random set Z in the extended convex ring in .d. For k . {1,.,d-1}, connections are shown between and the mean of . Here, the mean is understood to be with respect to the random isotropic k-dimensional linear subspace .k and the mean normal measure of the intersection is computed in .k. This mean to be well defined, a suitable spherical lifting must be applied to before averaging. A large class of liftings and their resulting means are discussed. In particular, a geometrically motivated lifting is presented, for which the mean of liftings of determines uniquely for any fixed k . {2,.,d-1}.