From the Hahn–Banach extension theorem to the isotonicity of convex functions and the majorization theory

@article{Niculescu2020FromTH,
  title={From the Hahn–Banach extension theorem to the isotonicity of convex functions and the majorization theory},
  author={Constantin P. Niculescu and Octav Olteanu},
  journal={Revista de la Real Academia de Ciencias Exactas, F{\'i}sicas y Naturales. Serie A. Matem{\'a}ticas},
  year={2020}
}
The property of isotonicity of a continuous convex function on the positive cone is characterized via subdifferentials. This is used to illustrate a new generalization of the Hardy–Littlewood–Polya inequality of majorization to the case of functions of a vector variable. 

The Hardy-Littlewood-Polya inequality of majorization in the context of w-m-star-convex functions

The Hardy-Littlewood-Polya inequality of majorization is extended for the {\omega}-m-star-convex functions to the framework of ordered Banach spaces. Several open problems which seem of interest for

On Special Properties for Continuous Convex Operators and Related Linear Operators

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric

Korovkin type theorems for weakly nonlinear and monotone operators

. In this paper we prove analogues of Korovkin’s theorem in the context of weakly nonlinear and monotone operators acting on Banach lattices of functions of several variables. Our results concern the

Nonlinear versions of Korovkin’s abstract theorems

In this paper we prove Korovkin type theorems for sequences of sublinear, monotone and weak additive operators acting on function spaces C ( X ), where X is a compact or a locally compact metric

Choquet operators associated to vector capacities

On Hahn-Banach theorem and some of its applications

Abstract First, this work provides an overview of some of the Hahn-Banach type theorems. Of note, some of these extension results for linear operators found recent applications to isotonicity of

From Hahn–Banach Type Theorems to the Markov Moment Problem, Sandwich Theorems and Further Applications

The aim of this review paper is to recall known solutions for two Markov moment problems, which can be formulated as Hahn–Banach extension theorems, in order to emphasize their relationship with the

On Markov Moment Problem and Related Results

TLDR
New results and theorems on the vector-valued Markov moment problem are proved by means of polynomial approximation on unbounded subsets, also applying an extension of the positive linear operators’ result.

Polynomial Approximation on Unbounded Subsets, Markov Moment Problem and Other Applications

This paper starts by recalling the author’s results on polynomial approximation over a Cartesian product A of closed unbounded intervals and its applications to solving Markov moment problems. Under

References

SHOWING 1-10 OF 29 REFERENCES

A Nonlinear Extension of Korovkin’s Theorem

In this paper we extend the classical Korovkin theorems to the framework of comonotone additive, sublinear and monotone operators. Based on the theory of Choquet capacities, several concrete examples

General inequalities via isotonic subadditive functionals

In this manuscript a number of general inequalities for isotonic subadditive functionals on a set of positive mappings are proved and applied. In particular, it is pointed out that these inequaliti

A note on the Choquet type operators

In this note the Choquet type operators are introduced, in connection to Choquet's theory of integrability with respect to a not necessarily additive set function. Based on their properties, a

Convex Functions, Monotone Operators and Differentiability

These notes start with an introduction to the differentiability of convex functions on Banach spaces, leading to the study of Asplund spaces and their intriguing relationship to monotone operators

Convex Functions: Constructions, Characterizations and Counterexamples

Preface 1. Why convex? 2. Convex functions on Euclidean spaces 3. Finer structure of Euclidean spaces 4. Convex functions on Banach spaces 5. Duality between smoothness and strict convexity 6.

Non-additive measure and integral

Preface. 1. Integration of Monotone Functions on Intervals. 2. Set Functions and Caratheodory Measurability. 3. Construction of Measures using Topology. 4. Distribution Functions, Measurability and

SUBLINEAR OPERATORS AND THEIR APPLICATIONS

ContentsIntroduction ??1. Sublinear operators ??2. Application of sublinear operators to the study of semigroups ??3. Superlinear point-set mappingsReferences

Convex analysis in general vector spaces

Preliminary Results on Functional Analysis Convex Analysis in Locally Convex Spaces Some Results and Applications of Convex Analysis in Normed Spaces.

Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces

This paper gives a survey over some of the most important methods and results of nonlinear functional analysis in ordered Banach spaces. By means of iterative techniques and by using topological

Generalized Measure Theory

TLDR
This chapter discusses the development of Structural Characteristics for Set Functions and Measurable Functions on Monotone Measure Spaces in the context of Generalized Measure Theory.