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aequationes mathematicae

, Volume 38, Issue 2–3, pp 192–210 | Cite as

On the lower hull of convex functions

  • Z. Kominek
  • M. Kuczma
Research Papers

Summary

Let (X, ℱ) be a topological space. For any functionf: D→[− ∞, ∞) (whereD ⊂ X), thelower hull m f :D →[− ∞, ∞) off is defined by
$$m_f (x) = m_{f\left| T \right.} (x) = \mathop {\sup \inf }\limits_{U \in T_x \in U \cap D} f(t),x \in D,$$
where ℱx denotes the family of all open sets containing x. The main result of the paper is that, ifX is a real linear topological Baire space,D ⊂ X is convex and open, andf: D→[− ∞, ∞) isJ-convex, then the functionm 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 letX be a real linear space. A setG ⊂ X is calledalgebraically open if for everyx ∈ G andy ∈ X there exists anε = ε(x, y) > 0 such thatx + λy ∈ G for λ ∈(−ε, ε). The family ℱ (X) of all algebraically open subsets ofX is a topology inX, which, however, is not linear (unless dimX = 1). For any functionf: D →[− ∞, ∞) thealgebraic lower hull m f * :D →[− ∞, ∞) is defined asm f * =m f|ℱ(x) . Again, ifD is convex and open andf isJ-convex, then the functionm f * is convex and continuous with respect to the topology ℱ(X).

IfX is a real linear topological space,D ⊂ X is convex and open, andf: D →[− ∞, ∞) is an arbitrary function, then bothm f andm f * are well defined inD. We always havem f ⩽ m f * ⩽ f; moreover,m f * =f wheneverf is convex, andm f * =m f wheneverf isJ-convex and dimX is finite, but in general neither of these equalities holds.

A number of related questions are also discussed. In particular, it is shown that, ifX is a real linear topological space,D ⊂ X is convex and open, andf: D →[− ∞, ∞) is aJ-convex function which is lower semicontinuous at every point of a setS ⊂ D containing a second category Baire subset, thenf is convex and continuous.

AMS (1980) subject classification

Primary 26A51 Secondary 39C05 

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Copyright information

© Birkhäuser Verlag 1989

Authors and Affiliations

  • Z. Kominek
    • 1
  • M. Kuczma
    • 1
  1. 1.Department of MathematicsSilesian UniversityKatowicePoland

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