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In this paper, we investigate properties of set-valued mappings thatestablish connection between the values of this map at two arbitrary points ofthe domain and the value at their midpoint. Such properties are, for instance,Jensen convexity/concavity, $K$-Jensen convexity/concavity (where $K$ is theset of nonnegative elements of an ordered vector space), and approximate/strong$K$-Jensen convexity/concavity. Assuming weak but natural regularityassumptions on the set-valued map, our main purpose is to deduce theconvexity/concavity consequences of these properties in the appropriate sense.Our two main theorems will generalize most of the known results in this field,in particular the celebrated Bernstein--Doetsch Theorem from 1915, and thusthey offer a unified view of these theories.

Bernstein--Doetsch type theorems for set-valued maps of strongly and approximately convex and concave type