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Continuity properties of convex-type set-valued maps. (English) Zbl 1061.26023
This paper offers an exhaustive survey on the relation between $$K$$-(mid-)convexity and $$K$$-(upper and lower semi-)continuity of set-valued maps acting between topological linear spaces. A group of results extends the classical results that concern the connection between convexity – continuity and midconvexity – local upper boundedness theorem (Bernstein-Doetsch). The closed graph theorem due to Robinson and Ursescu is also covered. Another group of results concerns superposition theorems. Finally, the last group of results is related to convexity with variable coefficients.

##### MSC:
 26E25 Set-valued functions 26A51 Convexity of real functions in one variable, generalizations 54C60 Set-valued maps in general topology 39B62 Functional inequalities, including subadditivity, convexity, etc.
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