Convex functions in linear spaces.

*(English)*Zbl 0699.46005
Pr. Nauk. Uniw. Śląsk. Katowicach 1087, 70 p. (1989).

Let X be a linear space and let D be a convex subset of X. A function f: \(D\to [-\infty,\infty)\) is called J-convex if it satisfies Jensen’s inequality \(f((x+y)/2)\leq (f(x)+f(y))/2\) for all \(x,y\in D\). A function f: \(X\to {\mathbb{R}}\) is called additive if it satisfies Cauchy’s functional equation \(f(x+y)=f(x)+f(y)\) for all \(x,y\in X.\)

This paper is concerned with properties of additive and convex functions defined on an open subset of a real linear space X endowed with a topology \(\tau\). Particularly important is the problem of the continuity of such functions. In this connection the following set classes are studied:

\(A(X,\tau)= \) \(\{\) \(T\subset X:\) every J-convex real function defined on an open and convex subset \(D\supset T\) of X which is bounded above on T, is continuous\(\}\)

\(B(X,\tau)= \) \(\{\) \(T\subset X:\) every additive function f: \(X\to {\mathbb{R}}\) which is bounded above on T is continuous\(\}\)

The theorems of Bernstein-Doetsch, Mehdi and Piccard are discussed and very general versions of them are presented. Convex functions in the sense of Wright are also considered and their connections with J-convex functions are studied.

Most results presented here have been published in the author’s earlier articles or in articles written jointly with R. Ger or M. Kuczma. In the present paper, however, they are proved under slightly weaker conditions and a more uniform presentation is given.

This paper is concerned with properties of additive and convex functions defined on an open subset of a real linear space X endowed with a topology \(\tau\). Particularly important is the problem of the continuity of such functions. In this connection the following set classes are studied:

\(A(X,\tau)= \) \(\{\) \(T\subset X:\) every J-convex real function defined on an open and convex subset \(D\supset T\) of X which is bounded above on T, is continuous\(\}\)

\(B(X,\tau)= \) \(\{\) \(T\subset X:\) every additive function f: \(X\to {\mathbb{R}}\) which is bounded above on T is continuous\(\}\)

The theorems of Bernstein-Doetsch, Mehdi and Piccard are discussed and very general versions of them are presented. Convex functions in the sense of Wright are also considered and their connections with J-convex functions are studied.

Most results presented here have been published in the author’s earlier articles or in articles written jointly with R. Ger or M. Kuczma. In the present paper, however, they are proved under slightly weaker conditions and a more uniform presentation is given.

Reviewer: I.Raşa

##### MSC:

46A55 | Convex sets in topological linear spaces; Choquet theory |

46G05 | Derivatives of functions in infinite-dimensional spaces |

26A51 | Convexity of real functions in one variable, generalizations |