Journal of Convex Analysis 26 (2019), No. 2, 563--572
Copyright Heldermann Verlag 2019
K-Midconvex and K-Midconcave Set-Valued Maps Bounded on "Large" Sets
Institute of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30-084 Krakow, Poland
Dept. of Mathematics, University of Bielsko-Biala, ul. Willowa 2, 43-309 Bielsko-Biala, Poland
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)].
Keywords: K-midconvex set-valued map, K-midconcave set-valued map, K-continuity, null-finite set, Haar-null set, Haar-meager set.
MSC: 26B25; 39B62, 54C60
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