On Weighted Parallel Volumes

The Wills functional of a convex body was originally defined as the sum of its intrinsic volumes. Meanwhile, various integral representations of the Wills functional have been introduced. In this paper we will introduce and examine the weighted parallel volumes as a class of functionals generalising the integral representations of the Wills functional. We will discuss to which extent the weighted parallel volumes are the linear combinations of intrinsic volumes and vice versa. The weighted parallel volumes can be considered as functionals defined on the set of all compact sets. We will study their properties and characterise the weighted parallel volumes which are continuous, additive resp. submodular. We will obtain most of our results in unsymmetric Minkowski spaces. Finally we apply some of our results to the capacity functional of Boolean models.