Related papers: Affine invariant valuations on polytopes
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…
A complete classification of all zonal, continuous, and translation invariant valuations on convex bodies is established. The valuations obtained are expressed as principal value integrals with respect to the area measures. The convergence…
A complete classification is established of Minkowski valuations on lattice polytopes that intertwine the special linear group over the integers and are translation invariant. In the contravariant case, the only such valuations are…
All continuous SL(n)-covariant $L_p$-Minkowski valuations defined on convex bodies are completely classified. The $L_p$-moment body operators turn out to be the nontrivial prototypes of such maps.
A characterization of valuations on the space of convex Lipschitz functions whose domain is a polytope in $\mathbb{R}^n$ is obtained. It is shown that every upper semicontinuous, equi-affine and dually epi-translation invariant valuation…
We classify all continuous valuations on the space of finite convex functions with values in the same space which are dually epi-translation-invariant and equi- resp. contravariant with respect to volume-preserving linear maps. We thereby…
Valuations on the space of finite-valued convex functions on $\mathbb{C}^n$ that are continuous, dually epi-translation invariant, as well as $\mathrm{U}(n)$-invariant are completely classified. It is shown that the space of these…
We shown that every continuous local functional on the space of finite convex functions on $\mathbb{R}^n$ is a valuation. This relation is used to establish a homogeneous decomposition for the class of polynomial local functionals as well…
Classifications of $\rm{SL}(n)$ covariant function-valued valuations are established with some assumptions of continuity. New valuations, for example, weighted moment functions, are introduced and our classifications give unified…
We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that…
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 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…
A classification of upper semicontinuous, translation and dually epi-translation invariant valuations is established on the space of convex Lipschitz function on $\mathbb{R}$ with compact domain.
The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree $n$ are…
Translation-invariant valuations on the space $L^\infty(\mathbb{R}^n)$ are examined. We prove that such functionals vanish on functions with compact support. Moreover a rich family of non-trivial translation-invariant valuations on…
We construct valuations on the space of finite-valued convex functions using integration of differential forms over the differential cycle associated to a convex function. We describe the kernel of this procedure and show that the…
The continuity of the inverse Klain map is investigated and the class of centrally symmetric convex bodies at which every valuation depends continuously on its Klain function is characterized. Among several applications, it is shown that…
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…
The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on…
In this paper, we endow the space of continuous translation invariant valuation on convex sets generated by mixed volumes coupled with a suitable Radon measure on tuples of convex bodies with two appropriate norms. This enables us to…