Related papers: Formulas for translative functions
Extended real-valued functions are often used in optimization theory, but in different ways for infimum problems and for supremum problems. We present an approach to extended real-valued functions that works for all types of problems and…
An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this,…
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.
We formulate a multi-valued version of the Tietze-Urysohn extension theorem. Precisely, we prove that any upper semicontinuous multi-valued map with nonempty closed convex values defined on a closed subset (resp. closed perfectly normal…
This paper investigates functions from $\mathbb{R}^d$ to $\mathbb{R} \cup \{\pm \infty\}$ that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on $d$, and…
Over the past years a theory of conjugate duality for set-valued functions that map into the set of upper closed subsets of a preordered topological vector space was developed. For scalar duality theory, continuity of convex functions plays…
This habilitation thesis centres on linearisation of vector-valued functions which means that vector-valued functions are represented by continuous linear operators. The first question we face is which vector-valued functions may be…
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 introduce vectorial and topological continuities for functions defined on vector metric spaces and illustrate spaces of such functions. Also, we describe some fundamental classes of vector valued functions and extension theorems.
A new directional derivative and a new subdifferential for set-valued convex functions are constructed, and a set-valued version of the so-called 'max-formula' is proven. The new concepts are used to characterize solutions of convex…
In this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if $L$ is a subfit frame, arbitrary extended real functions on $L$ are the elements of the…
Generalized eigenfunctions may be regarded as vectors of a basis in a particular direct integral of Hilbert spaces or as elements of the antidual space $\Phi^\times$ in a convenient Gelfand triplet…
We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type…
All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified.
Functions with uniform sublevel sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used in multicriteria optimization, decision theory, mathematical…
A complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on ${\mathbb R}^n$ is established. The valuations obtained are functional versions of the…
We introduce real vector spaces composed of set-valued maps on an open set. They are also complete metric spaces, lattices, commutative rings. The set of differentiable functions is a dense subset of these spaces and the classical gradient…
The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…
Given a strictly positive measure, we characterize inner semicontinuous solid convex-valued mappings for which continuous functions which are selections almost everywhere are selections. This class contains continuous mappings as well as…
We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of…