A generalization of intersection formulae of integral geometry
The paper concerns generalizations of two intersection formulae for curvature measures of convex bodies: the principal kinematic formula and the Crofton formula (Theorems 1.1 and 1.2). The author proves two pairs of theorems: \{3.1, 3.2\} and \{3.3, 3.4\}. All of these theorems deal with the so-called support measures (generalized curvature measures), which are due to R. Schneider. The first pair concerns convex bodies, the second concerns elements of convex rings, i.e. finite unions of convex bodies. Theorems 3.1 and 3.3 are counterparts of the principal kinematic formula; they are proved under some additional assumption (for the so-called admissible convex bodies and, respectively, admissible elements of the convex ring). It is an open problem whether this assumption is essential. Theorems 3.2 and 3.4 are generalizations of the Crofton formula.
scientific article; zbMATH DE number 1098494