Journal of Convex Analysis 28 (2021), No. 1, 103--122
Copyright Heldermann Verlag 2021
Spherically Convex Sets and Spherically Convex Functions
Qi Guo
Department of Mathematics, Suzhou University of Science and Technology, 215009 Suzhou, Jiangsu, China
guoqi@mail.usts.edu.cn
Yanling Peng
Department of Mathematics, Suzhou University of Science and Technology, 215009 Suzhou, Jiangsu, China
yanlingpeng35@hotmail.com
We define first the spherical convexity of sets and functions on general curved surfaces by an analytic approach. Then we study several kinds of properties of spherically convex sets and functions. Several analogies of the results for convex sets and convex functions on Euclidean spaces are established or rediscovered for spherically convex sets and spherically convex functions, such as the Radon-type, Helly-type, Carathéodory-type and Minkowski-type theorems for spherically convex sets, and the Jensen's inequality for spherically convex functions etc. The results obtained here might have applications in some areas, e.g. in the optimization theory on general spherical spaces.
Keywords: Spherical convexity, Helly theorem, Carathéodory theorem, Jensen's inequality.
MSC: 52A55, 52A20, 52A35, 52A41.
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