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

Abstract

Letbe the mean normal measure of a stationary random setZin the extended convex ring in ℝd. Fork∈ {1,…,d-1}, connections are shown betweenand the mean of. Here, the mean is understood to be with respect to the random isotropick-dimensional linear subspace ξkand the mean normal measure of the intersection is computed in ξk. This mean to be well defined, a suitable spherical lifting must be applied tobefore 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 ofdeterminesuniquely for any fixedk∈ {2,…,d-1}.