On the extension of additive functionals on classes of convex sets

Curvature measures and fractals

“…This leads to the additive extension of curvature measures to the convex ring. Groemer showed in [12] that such an extension is indeed possible and unique, i.e. the so defined measures C k (K, · ) do not depend on the chosen representation of K ∈ R d by convex sets K i .…”

“…Here the additivity of the curvature measures for sets with positive reach is used to define them for sets which can be represented as (locally finite) unions of these sets. Such an extension has first been considered by Groemer [12] for the subclass K d of compact, convex sets, introducing curvature measures for polyconvex sets, i.e. finite unions of convex sets, in this way.…”

Choquet–Kendall–Matheron theorems for non-Hausdorff spaces

“…We will always suppose that is a ∩-semilattice, that is, finite intersections of members of also belong to . Under this hypothesis Groemer (Groemer, 1978, Theorem 1) has given a complete solution to the problem. We refer to the book by Klain and Rota for an elegant presentation of Groemer's Integral Theorem (Klain and Rota, 1997, Theorem 2.2.1).…”

Integral Geometric Tools for Stochastic Geometry

“…In adopting the name 'polyconvex' for the elements of the convex ring, we followed Klain and Rota [25], who in turn followed E. de Giorgi. The extension theorem 2.5 and its proof reproduced here are due to Groemer [10]. For the support measures, and thus for the curvature measures, a more explicit construction of an additive extension to polyconvex sets is found in Section 4.4 of [39].…”

“…. , V n , uniquely determined by (10), are called the intrinsic volumes (also, with different normalizations, the quermassintegrals or Minkowski functionals). About their geometric meaning, the following can be said.…”