Abstract

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.

Author information

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

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

Keywords

  • Spherical convexity
  • Helly theorem
  • Carathéodory theorem
  • Jensen's inequality

Mathematics Subject Classification

52A55, 52A20, 52A35, 52A41

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Suggested citation

Q. Guo, Y. Peng. “Spherically Convex Sets and Spherically Convex Functions.” Journal of Convex Analysis 28 (2021), No. 1, 103–122. https://doi.org/10.68381/jca28009

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Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.