Geodesic
K-Midconvex and K-Midconcave Set-Valued Maps Bounded on "Large" Sets
Journal of convex analysis, Tome 26 (2019) no. 2, pp. 563-572
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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)].
Classification : 26B25, 39B62, 54C60
Mots-clés : K-midconvex set-valued map, K-midconcave set-valued map, K-continuity, null-finite set, Haar-null set, Haar-meager set 
@article{JCA_2019_26_2_JCA_2019_26_2_a9,
     author = {E. Jablonska and K. Nikodem},
     title = {K-Midconvex and {K-Midconcave} {Set-Valued} {Maps} {Bounded} on {"Large"} {Sets}},
     journal = {Journal of convex analysis},
     pages = {563--572},
     year = {2019},
     volume = {26},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2019_26_2_JCA_2019_26_2_a9/}
}
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E. Jablonska; K. Nikodem. K-Midconvex and K-Midconcave Set-Valued Maps Bounded on "Large" Sets. Journal of convex analysis, Tome 26 (2019) no. 2, pp. 563-572. http://geodesic.mathdoc.fr/item/JCA_2019_26_2_JCA_2019_26_2_a9/