Related papers: The Klain approach to zonal valuations

A functional analog of the Klain-Schneider theorem for vector-valued valuations on convex functions is established, providing a classification of continuous, translation covariant, simple valuations. Under additional rotation equivariance…

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…

We provide a new proof of Alesker's Irreducibility Theorem. We first introduce a new localization technique for polynomial valuations on convex bodies, which we use to independently prove that smooth and translation invariant valuations are…

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…

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…

New proofs of the Hadwiger theorem for smooth and for continuous valuations on convex functions are obtained, and the Klain-Schneider theorem on convex functions is established. In addition, an extension theorem for valuations defined on…

This article is concerned with the representation of curves by means of integral invariants. In contrast to the classical differential invariants they have the advantage of being less sensitive with respect to noise. The integral invariant…

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 propose a new inner product for scalar fields that are solutions of the Klein-Gordon equation with $m^2<0$. This inner product is non-local, bearing an integral kernel including Bessel functions of the second kind, and the associated…

The famous Hadwiger theorem classifies all rigid motion invariant continuous valuations on convex sets as linear conbinations of quermassintegrals. We prove much more general result. We classify continuous valuations which are invariant…

We study in detail certain natural continuous representations of G = GL(n,K) in locally convex vector spaces over a locally compact, non-archimedean field K of characteristic zero. We construct boundary value maps, or integral transforms,…

For valuations on convex bodies in Euclidean spaces, there is by now a long series of characterization and classification theorems. The classical template is Hadwiger's theorem, saying that every rigid motion invariant, continuous,…

We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are…

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…

In this paper we construct a geometric analogue of the Weil representation over a finite field. Our construction is principally invariant, not choosing any specific realization. This eliminates most of the unpleasant formulas that appear in…

We compute the value of finitary localizing invariants, including algebraic $K$-theory, on categories of sheaves over stably locally compact spaces $X$. Our formula simultaneously generalizes the cases of locally compact Hausdorff and…

We investigate the invariant metrics and complex geodesics in the universal Teichm\"{u}ller space and Teichm\"{u}ller space of the punctured disk using Milin's coefficient inequalities. This technique allows us to establish that all…

We aim addition theorems for multivariate Krawtchouk polynomials, following Dunkl(1976) for 1-variate case. We work on harmonic analysis on a non-Archimedean local field, that is a group theoretic situation where these polynomials play…

In this paper, we attempt to shed light on a new class of nonstationary random fields which exhibit, what we call, local invariant nonstationarity. We argue that the local invariant property has a special interaction with a new generalized…

For an additive polynomial and a positive integer, we define an irreducible smooth representation of a Weil group of a non-archimedean local field. We study several invariants of this representation. We deduce a necessary and sufficient…