Integral geometry of translation invariant functionals. II: The case of general convex bodies (Q343839)

The author continues his study of translation invariant functionals, which was started in [Adv. Appl. Math. 66, 46--79 (2015; Zbl 1327.52008)]. In contrast to the first part of this study in which translation invariant functionals on convex polytopes were investigated, the author discusses now such functionals on the set \(\mathcal{K}\) of all convex bodies together with the corresponding translative integral formulas. Local extensions are again considered and used to show that the translative formulas extend to arbitrary continuous and translation invariant valuations. Furthermore, kinematic formulas are studied and certain results from the first part are extended to mean values of valuations for Boolean models with general convex or polyconvex grains. The paper is concluded with the discussion of special cases leading to new integral and mean value formulas for flag measures.