On the lower hull of convex functions

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 ℱ_x 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.