Sui concetti di K-convessità (K-concavità) e di K-convessità* (K-concavità*)

This note is connected with some results of K. Nikodem on the relations between midpoint-K-convexity and K-continuity for set-valued functions on linear topological spaces. The authors prove that if K is closed and F(x) is compact for every x in D, F being midpoint K-convex and K-continuous on D, then F is K-convex. Moreover, if X = R^n, F is K-convex and F(x) bounded for every x in D, then F is K-continuous. The second part of the research studies the concepts of K-convexity [K-concavity] and those of midpoint K-convexity [Kconcavity].