# On the lower hull of convex functions

- 66 Downloads

## Summary

*X, ℱ*) be a topological space. For any function

*f: D→[− ∞, ∞)*(where

*D ⊂ X*), the

*lower hull m*

_{ f }:

*D →[− ∞, ∞)*of

*f*is defined by

_{x}denotes the family of all open sets containing x. The main result of the paper is that, if

*X*is a real linear topological Baire space,

*D ⊂ X*is convex and open, and

*f: D→[− ∞, ∞)*is

*J*-convex, then the function

*m*

_{ f }is convex and continuous. (In the case of a single real variable this result goes back to F. Bernstein and G. Doetsch, 1915.)

Now let*X* be a real linear space. A set*G ⊂ X* is called*algebraically open* if for every*x ∈ G* and*y ∈ X* there exists an*ε = ε(x, y)* > 0 such that*x + λy ∈ G* for λ ∈(−ε, ε). The family ℱ (*X*) of all algebraically open subsets of*X* is a topology in*X*, which, however, is not linear (unless dim*X* = 1). For any function*f: D →[− ∞, ∞)* the*algebraic lower hull m* _{ f } ^{*} :*D →[− ∞, ∞)* is defined as*m* _{ f } ^{*} =*m*_{ f|ℱ(x) }. Again, if*D* is convex and open and*f* is*J*-convex, then the function*m* _{ f } ^{*} is convex and continuous with respect to the topology ℱ*(X).*

If*X* is a real linear topological space,*D ⊂ X* is convex and open, and*f: D →[− ∞, ∞)* is an arbitrary function, then both*m*_{ f } and*m* _{ f } ^{*} are well defined in*D.* We always have*m*_{ f }*⩽ m* _{f} ^{*} *⩽ f*; moreover,*m* _{f} ^{*} =*f* whenever*f* is convex, and*m* _{f} ^{*} =*m*_{ f } whenever*f* is*J*-convex and dim*X* is finite, but in general neither of these equalities holds.

A number of related questions are also discussed. In particular, it is shown that, if*X* is a real linear topological space,*D ⊂ X* is convex and open, and*f: D →[− ∞, ∞)* is a*J*-convex function which is lower semicontinuous at every point of a set*S ⊂ D* containing a second category Baire subset, then*f* is convex and continuous.

### AMS (1980) subject classification

Primary 26A51 Secondary 39C05## Preview

Unable to display preview. Download preview PDF.

### References

- [1]Barbu, V. and Precupanu, Th.,
*Convexity and optimization in Banach spaces.*Editura Academiei Bucureşti, and Sijthoff & Nordhoff International Publishers, Alphen aan den Rijn, 1978.Google Scholar - [2]Bernstein, F. undDoetsch, G.,
*Zur Theorie der konvexen Funktionen*. Math. Ann.*76*(1915), 514–526.CrossRefGoogle Scholar - [3]Kominek, B. andKominek, Z.,
*On some set classes connected with the continuity of additive and Q-convex functions*. Prace Nauk. Uniw. Śląsk.*8*(1978), 60–63.Google Scholar - [4]Kominek, Z.,
*On the continuity of Q-convex and additive functions*. Aequationes Math.*23*(1981), 146–150.Google Scholar - [5]
- [6]Kuczma, M.,
*An introduction to the theory of functional equation and inequalities. Cauchy's equation and Jensen's inequality.*PWN, Warszawa—Kraków—Katowice, 1985.Google Scholar - [7]Kuczma, M.,
*On some analogies between measure and category and their applications in the theory of additive functions*. Ann. Math. Sil.*1*(1985), 155–162.Google Scholar - [8]Kuhn, N.,
*A note on t-convex functions.*In General Inequalities 4 [Proceedings of the Fourth International Conference on General Inequalities held in the Mathematical Institute at Oberwolfach, Black Forest, May 8–14, 1983]. Edited by W. Walter, International Ser. Numer. Math. 71. Birkhäuser Verlag, Basel—Stuttgart, 1984, pp. 269–276.Google Scholar - [9]
- [10]
- [11]Roberts, A. W. andVarberg, D. E.,
*Convex functions*. Academic Press, New York—London, 1973.Google Scholar - [12]Valentine, F. A.,
*Convex sets*. McGraw-Hill Book Company, New York—San Francisco—Toronto—London, 1964.Google Scholar