Geodesic
Convexity with Respect to Beckenbach Families
Journal of convex analysis, Tome 24 (2017) no. 1, pp. 75-92
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Beckenbach families are sets of functions possessing the two characteristic properties of Euclidean lines: their members are continuous and each distinct pairs of points of the plane can be interpolated by a unique member of the family. Applying Beckenbach families, the notion of (planar) convexity can be extended. Moreover, generalized convex functions can also be studied in this framework. The aim of this note is to prove the analogue of the Radon, Helly, Carathéodory and Minkowski Theorems in this generalized setting. The most important properties of generalized convex functions are presented, as well. As applications, some separation results are given.
Classification : 52A10, 26A51, 39B62, 52A40
Mots-clés : Beckenbach families, convex sets and functions, separation theorems 
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     author = {M. Bessenyei and A. Konkoly and B. Popovics},
     title = {Convexity with {Respect} to {Beckenbach} {Families}},
     journal = {Journal of convex analysis},
     pages = {75--92},
     year = {2017},
     volume = {24},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2017_24_1_JCA_2017_24_1_a5/}
}
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M. Bessenyei; A. Konkoly; B. Popovics. Convexity with Respect to Beckenbach Families. Journal of convex analysis, Tome 24 (2017) no. 1, pp. 75-92. http://geodesic.mathdoc.fr/item/JCA_2017_24_1_JCA_2017_24_1_a5/