Translative and kinematic integral formulae concerning the convex hull operation
نویسنده
چکیده
exists and can be expressed in terms of K and K ′. 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.
منابع مشابه
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 functiona...
متن کامل2 KINEMATIC FORMULAE FOR SUPPORT MEASURES OF CONVEX BODIESif
A new kinematic formula for the support measures of convex bodies is proved. The underlying operation is the convex hull operation for pairs of convex bodies. Further, the concept of mixed support measures is introduced and kinematic relations for these new functionals are indicated.
متن کاملIntersections and translative integral formulas for boundaries of convex bodies
Let K,L ⊂ IRn be two convex bodies with non-empty interiors and with boundaries ∂K, ∂L, and let χ denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t 7→ χ(∂K ∩ (∂L + t)) and t 7→ χ(∂K ∩ (L + t)), t ∈ IRn, with respect to an ndimensional Haar mea...
متن کاملIntersection Formulae of Integral Geometry 21
We establish extensions of the Crofton formula and, under some restrictions, of the principal kinematic formula of integral geometry from curvature measures to generalized curvature measures of convex bodies. We also treat versions for nite unions of convex bodies. As a consequence, we get a new intuitive interpretation of the area measures of Aleksandrov and Fenchel{ Jessen. The subject of thi...
متن کاملIntegral Geometric Tools for Stochastic Geometry
Integral geometry, as it is understood here, deals with measures on sets of geometric objects, and in particular with the determination of the total measure of various such sets having geometric significance. For example, given two convex bodies in Euclidean space, what is the total invariant measure of the set of all rigid motions which bring the first set into a position where it has nonempty...
متن کامل