On Evenly Convex Sets and Functions

A subset of R^n is said to be evenly convex (e-convex, in breaf) if it is the intersection of some family (possibly empty) of open halfspaces. In this paper, we collect some published results which show that this large class of convex sets enjoys a lot of the well-known properties of the subclass of closed convex sets. We also consider functions whose epigraphs are e-convex sets, the so-called e-convex functions, and we show the main properties of this class of convex functions that contains the important class of lower semicontinuous convex functions.Esta publicación se ha realizado con el apoyo financiero del Ministerio de Economía y Competitividad, a través del proyecto MTM2011-29064-C03-02

On geodesic semi strongly E-convex functions

In this article, a new class of function called geodesic semi strongly E-convex functions and generalized geodesic semi strongly E-convex functions are introduced. Also, some of their properties are obtained

hh-strongly EE-convex functions

Starting from strongly $E$-convex functions introduced by E. A. Youness, and T. Emam, from $h$-convex functionsintroduced by S. Varošanec and from the more general conceptof $h$-convex functions introduced by A. Házy we define and study $h$-strongly $E$-convex functions. We study some properties of them

Stable and total Fenchel duality for convex optimization problems in locally convex spaces

We consider the optimization problem (PA) infx∈X{f(x) + g(Ax)} where f and g are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y, respectively, and A is a linear operator from X to Y . By using the properties of the epigraph of the conjugated functions, some sufficient and necessary conditions for the strong Fenchel duality and the strong converse Fenchel duality of (PA) are provided. Sufficient and necessary conditions for the stable Fenchel duality and for the total Fenchel duality are also derived.National Natural Science Foundation of ChinaDirección General de Enseñanza SuperiorMinisterio de Ciencia e Innovació

Fractional BV spaces and first applications to scalar conservation laws

The aim of this paper is to obtain new fine properties of entropy solutions of nonlinear scalar conservation laws. For this purpose, we study some "fractional $BV$ spaces" denoted $BV^s$, for $0 < s \leq 1$, introduced by Love and Young in 1937. The $BV^s(\R)$ spaces are very closed to the critical Sobolev space $W^{s,1/s}(\R)$. We investigate these spaces in relation with one-dimensional scalar conservation laws. $BV^s$ spaces allow to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with $BV^s$ initial data. Furthermore, for the first time we get the maximal $W^{s,p}$ smoothing effect conjectured by P.-L. Lions, B. Perthame and E. Tadmor for all nonlinear degenerate convex fluxes

Analysis of Convex Functions

Convexity is an old subject in mathematics. The �rst speci�c de�nition of convexity was given by Herman Minkowski in 1896. Convex functions were introduced by Jensen in 1905. The concept appeared intermittently through the centuries, but the subject was not really formalized until the seminal 1934 tract Theorie der konverxen Korper of Bonneson and Fenchel.Today convex geometry is a mathematical subject in its own right. Classically oriented treatments, like the work done by Frederick Valentine form the elementary de�nition, which is that a domain K in the plane or in RN is convex if for all P;Q 2 K, then the segment PQ connecting P to Q also lies in K. In fact this very simple idea gives forth a very rich theory. But it is not a theory that interacts naturally with mathematical analysis. For analysis, one would like a way to think about convexity that is expressed in the language of functions and perhaps its derivatives. Our goal in this thesis is to present and to study convexity in a more analytic way. Through Chapter 1, Chapter 2 and Chapter 3, I have tried to point out the important role of convex sets and its associated convex functions in Mathematical Analysis. Chapter 1 is devoted to Convex sets and some geometric properties achieved by these objects in �nite Euclidean spaces. The emphasis is given on establishing a criteria for convexity. Various useful examples are given, and it is shown how further examples can be generated from these by means of operations such as addition or taking convex hulls. The fundamental idea to be understood is that the convex functions on Rn can be identi�ed with certain convex subsets of Rn+1 (their epigraphs), while the convex sets in Rn can be identi�ed with certain convex functions on Rn (their indicators). These identi�cations make it easy to pass back and forth between a geometric approach and an analytic approach. Chapter 2 begins with idea of convexity of functions in a �nite dimensional space. Convex functions are an important device for the study of extremal problems. They are also important analytic tools. The fact that a convex function can have at most one minimum and no maxima is a notable piece of information that proves to be quite useful. A convex function is also characterized by the non negativity of its second derivative. This useful information interacts nicely with the ideas of calculus. We relate convex functions to an elegant characterisation of Gamma functions by Bohr Mollerup Theorem. Chapter 3 provides an introduction to convex analysis, the properties of sets and functions in in�nite dimensional space. We start by taking the convexity of the epigraph to be the definition of a convex function, and allow convex functions to be extended -real valued. One of the main themes of this chapter is the maximization of linear functions over non empty convex sets. Here we relate the subdi�erential to the directional derivative of a function. There are several modern works on convexity that arise from studies of functional analysis. One of the nice features of the analytic way of looking at convexity is the Bishop-Phelps Theorem, it says that in a Banach Space, a convex function has a subgradient on a dense subset of its e�ective