Related papers: An extension problem for convex functions

We show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems,…

Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…

We study the local and global versions of the convexity, which is closely related to the problem of extending a convex function on a non-convex domain to a convex function on the convex hull of the domain and beyond the convex hull. We also…

We provide proof that the optimal value function of a convex parametrized optimization problem in Euclidean spaces is itself a convex function onto the extended real line.

Some concepts, such as non-compactness measure and condensing operators, defined on metric spaces are extended to uniform spaces. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in…

We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave functions on such spaces satisfy several strong continuity properties.

We derive two types of representation results for increasing convex functionals in terms of countably additive measures. The first is a max-representation of functionals defined on spaces of real-valued continuous functions and the second a…

The minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces. Upper and lower bounds for the minimax risk can still be described in terms of a modulus of continuity. However in…

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…

We study a specific convex maximization problem in the space of continuous functions defined on a semi-infinite interval. An unexplained connection to the discrete version of this problem is investigated.

We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the…

Let $H$ be a real Hilbert space. In this short note, using some of the properties of bounded linear operators with closed range defined on $H$, certain bounds for a specific convex subset of the solution set of infinite linear…

We quickly review and make some comments on the concept of convexity in metric spaces due to Takahashi. Then we introduce a concept of convex structure based convexity to functions on these spaces and refer to it as $W-$convexity.…

This note deals with certain properties of convex functions. We provide results on the convexity of the set of minima of these functions, the behaviour of their subgradient set under restriction, and optimization of these functions over an…

There are two definitions of the measurable functional on the topological vector space: as a linear and measurable real-valued function and as a pointwise limit of the sequence of the continious linear functionals. In general case they are…

We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined…

We prove a general extension theorem for holomorphic line bundles on reduced complex spaces, equipped with singular hermitian metrics, whose curvature currents can be extended as positive, closed currents. The result has applications to…

It is shown that a possibly infinite-valued proper lower semicontinuous convex function on ${\mathbb R}^n$ has an extension to a convex function on the half-space ${\mathbb R}^n\times[0,\infty)$ which is finite and smooth on the open…

We consider scalar equilibrium problems governed by a bifunction in a finite-dimensional framework. By using classical arguments in Convex Analysis, we show that under suitable generalized convexity assumptions imposed on the bifunction,…

Let $E$ be an arbitrary subset of a Banach space $X$, $f: E \rightarrow \mathbb{R}$ be a function, and $G:E \rightrightarrows X^*$ be a set-valued mapping. We give necessary and sufficient conditions on $f, G$ for the existence of a…