In this interesting paper the author studies the problem of determining the (oriented) mean normal measure \(\overline{S}(Z,\cdot)\) [introduced by W. Weil, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 50, 387-412 (1997; Zbl 0891.60017)] of a stationary random closed set in the extended convex ring in \(\mathbb{R}^d\), which is a local counterpart of the surface area density. Starting from randomly chosen \(k\)-dimensional flat sections \(\xi_k\) \((k\in\{1,\dots, d-1\})\) of the set, he discusses different aspects of the determination of \(\overline{S}(Z,\cdot)\) from the mean normal measure of the intersection \(\mu_k=\overline{S}'(Z\cap \xi_k,\cdot)\), computed in \(\xi_k\).
Let \(\tau_{\xi_k}\mu_k\) denote the measure on \(\mathbb{S}^{d-1}\) that is supported by \(\mathbb{S}^{d-1}\cap \xi_k\) and equals \(\mu_k\) there. If \(\xi_k\) is assumed to be isotropic, then \(E_{\nu_k}(\tau_{\xi_k}\mu_k)=\pi_k\overline{S}(Z,\cdot)\). This gives the expectation with respect to \(\nu_k\), the rotation-invariant probability measure on the space of \(k\)-dimensional linear subspaces, in terms of \(\pi_k\), which is a weakly continuous linear transformation of the space of signed measures into itself [see P. Goodey, M. Kiderlen and W. Weil, Monatsh. Math. 126, 37-54 (1998; Zbl 0922.52001)]. The stereological injectivity problem for \(\pi_k\) is generalized by replacing the trivial lifting \(\tau_{\xi_k}\) with more general spherical liftings, and properties of the average depending on the lifting are derived. In particular, a geometrically motivated lifting is presented, for which, for any fixed \(k\in\{2,\dots, d-1\}\), the mean of liftings of \(\overline{S}'(Z\cap \xi_k,\cdot)\) determines \(\overline{S}(Z,\cdot)\) uniquely.