Rotational Crofton formulae for Minkowski tensors and some affine counterparts (Q2399886)

The classical Crofton formula in \(\mathbb R^d\) states that \[ \int_{\mathcal{E}_j^d} V_k(K \cap E) dE = V_{d+k-j}(K), \] where \(\mathcal{E}_j^d\) denotes the affine Grassmann manifold endowed with some translation- and rotation-invariant measure, \(K\) is a compact convex body and \(V_k\) is the \(k\)-th intrinsic volume. It can be extended to compact sets with positive reach, i.e., sets \(X\) such that there exists some \(r>0\) with the property that each point at distance less than \(r\) to \(X\) has a unique nearest point on \(X\). In the present paper, a Crofton-type formula for Minkowski tensors is proved. Minkowski tensors are valuations on compact sets or sets with positive reach which take their values in the space of symmetric tensors. They contain more information about the set as the usual intrinsic volumes, in particular about the spatial position of the set. The main result is a rotational Crofton formula for Minkowski tensors, where the affine Grassmannian is replaced by the linear Grassmannian. The integral is then expressed by an explicit formula involving the normal cycle of \(X\) and generalized curvature measures of \(X\). The motivation for such formulas comes from local stereology. The cases which are of a particular interest in these applications are studied in greater detail.