K-Midconvex and K-Midconcave Set-Valued Maps Bounded on "Large" Sets

We prove that every K-midconvex set-valued map partially K-upper bounded on a "large" set (e.g. not null-finite, not Haar-null or not Haar-meager set), as well as every K-midconcave set-valued map K-lower bounded on a "large" set, is K-continuous. These results generalize the solution of the problem posed by K. Baron and R. Ger [Problem P239, in: The 21st International Symposium on Functional Equations, Konolfingen (1983), Aequationes Math. 26 (1984) 225--294], and also some results of K. Nikodem [K-convex and K-concave set-valued functions, Zeszyty Nauk. Politechniki Lodzkiej Mat. 559; Rozprawy Mat. 114, Lodz (1989)].