On the lower hull of convex functions
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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 classificationPrimary 26A51 Secondary 39C05
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