This note starts from recent research of K. Nikodem [Aequationes Math. 33 (1987), no. 1, 46– 56; MR0901799 (88h:90171)] on the continuity of convex multifunctions, whose results are extended. Here the authors introduce the concept of midpoint convexity for multifunctions and give conditions for lower semicontinuity. A similar theorem holds for midpoint concavity. From these theorems there follows a condition of continuity for midpoint additive multifunctions.
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