Abstract

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 Łódzkiej Mat. 559; Rozprawy Mat. 114, Łódź (1989)].

Author information

Contact details are reproduced from the original publication and may be historical.

Eliza Jabłońska

Institute of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30-084 Krakow, Poland

elizajab0712@gmail.com

Kazimierz Nikodem

Dept. of Mathematics, University of Bielsko-Biala, ul. Willowa 2, 43-309 Bielsko-Biała, Poland

knikodem@ath.bielsko.pl

Keywords

  • K-midconvex set-valued map
  • K-midconcave set-valued map
  • K-continuity
  • null-finite set
  • Haar-null set
  • Haar-meager set

Mathematics Subject Classification

26B25; 39B62, 54C60

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Suggested citation

E. Jabłońska, K. Nikodem. “K-Midconvex and K-Midconcave Set-Valued Maps Bounded on "Large" Sets.” Journal of Convex Analysis 26 (2019), No. 2, 563–572. https://doi.org/10.68381/jca26029

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.