On Convex Functions with Values in Semi–linear Spaces
نویسنده
چکیده
The following result of convex analysis is well–known [2]: If the function f : X → [−∞, +∞] is convex and some x0 ∈ core (dom f) satisfies f(x0) > −∞, then f never takes the value −∞. From a corresponding theorem for convex functions with values in semi–linear spaces a variety of results is deduced, among them the mentioned theorem, a theorem of Deutsch and Singer on the single–valuedness of convex set–valued maps as well as a result on the compact–valuedness of convex set–valued maps. We also discuss the possibility of embedding the image points of such a convex function into a linear space.
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