Translative and Kinematic Integral Formulae concerning the Convex Hull Operation Translative and Kinematic Integral Formulae

For convex bodies K; K 0 and a translation in n-dimensional Euclidean space, let K _ K 0 be the convex hull of the union of K and K 0. Let F be a geometric functional on the space of all convex bodies. We consider special families (r) r>0 of measures on the translation group T n such that the limit lim r!1 Z Tn F (K _ K 0) dd r () exists and can be expressed in terms of K and K 0. The functionals F under consideration are derived from the mixed volume or the mixed area measure functional. Analogous questions are treated for the motion group instead of the translation group. The resulting relations can be regarded as \dual" counterparts to various versions of the principal kinematic formula. Motivation for our investigations is provided by classical and recent results from spherical integral geometry. The classical results of convexity-related integral geometry concern mean value formulae for intersections, orthogonal projections and Minkowski sums of convex bodies. One further basic operation, which is in some sense dual to the intersection, is deened by the convex hull of the union of two convex bodies. Integral formulae concerning this operation are known in spherical integral geometry, as intersection formulae can be transferred, due to the spherical principle of duality, into results regarding the convex hull operation. It is clear that, if there are corresponding formulae in Euclidean space, they must be of a diierent nature because of the non-compactness of Euclidean space and its motion group. It is shown in this article that there are kinematic formulae, in the form of limit relations, which resemble these spherical results. They have particularly nice properties with respect to Minkowski addition, and they are related to the well-known rotation sum and projection formulae. In a rst step, we prove translative versions, which have a simpler structure than the known translative intersection formulae. It is interesting that the invariant measure on the translation group (Lebesgue measure) can be replaced by much more general measures while still leading to simple explicit results. We then apply known rotational mean value formulae to deduce kinematic versions from our translative results. For integral geometry of spherically convex bodies, we refer to the thesis 3], which contains classical as well as new results; see also Part IV of Santall o's book 6] and the literature quoted there. For integral geometry …