Related papers: Generalized translation invariant valuations and t…

We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and…

We study real-valued, continuous and translation invariant valuations defined on the space of quasi-concave functions of N variables. In particular, we prove a homogeneous decomposition theorem of McMullen type, and we find a representation…

A classical theorem of P. McMullen describes all valuations on polytopes that are invariant under translations and weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically…

This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of a linear space. In this article we still work…

All simple translation-invariant valuations on polytopes are classified. As a direct consequence the well-known conditions for translative-equidecomposability are recovered. Furthermore, a simplified proof of the classification of…

We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was…

We study translation invariant, real-valued valuations on the class of convex polytopes in Euclidean space and discuss which continuity properties are sufficient for an extension of such valuations to all convex bodies. For this purpose, we…

A convolution representation of continuous translation invariant and SO(n) equivariant Minkowski valuations is established. This is based on a new classification of translation invariant generalized spherical valuations. As applications,…

Let $V$ be a finite dimensional real vector space. In the article we construct an isomorphism between the space of smooth translation invariant valuations on convex subsets of $V$ and the space of such valuations (twisted by densities) on…

We study translative integral formulas for certain translation invariant functionals on convex polytopes and discuss local extensions and applications to Poisson processes and Boolean models.

In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative…

We present elements of a theory of translation-invariant integration on finite dimensional vector spaces and on GL_n over a valuation field with local field as residue field. We then discuss the case of an arbitrary algebraic group. This…

We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of…

A classification of SL$(n)$ invariant valuations on the space of convex polytopes in $R^n$ without any continuity assumptions is established. A corresponding result is obtained on the space of convex polytopes in $R^n$ that contain the…

We study dually epi-translation invariant valuations on cones of convex functions containing the space of finite-valued convex functions. The existence of a homogeneous decomposition is used to associate a distribution to every valuation of…

In this paper we introduce and study a topological abelian group of convex bodies, analogous to the scissors congruence group and McMullen's polytope algebra, with the universal property that continuous valuations on convex bodies…

Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real orthogonal group with signature $(p,q)$. We give a complete description of the spaces of continuous and generalized translation- and $\mathrm{SO}^+(p,q)$-invariant…

McMullen's g-vector is important for simple convex polytopes. This paper postulates axioms for its extension to general convex polytopes. It also conjectures that, for each dimension d, a stated finite calculation gives the formula for the…

This is the fourth part in the series of articles math.MG/0503397, math.MG/0503399, math.MG/0509512 where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations introduced in…