New Proofs for some Results on Spherically Convex Sets

C. Zalinescu

Journal of convex analysis, Tome 29 (2022) no. 4, pp. 995-1006

Résumé

In Guo and Peng's article [Spherically convex sets and spherically convex functions, J. Convex Analysis 28 (2021) 103--122] the authors define the notions of spherical convex sets and functions on "general curved surfaces" in Rⁿ (n ≥ 2), they study several properties of these classes of sets and functions, and they establish analogues of Radon, Helly, Carathéodory and Minkowski theorems for spherical convex sets, as well as some properties of spherical convex functions which are analogous to those of usual convex functions. In obtaining such results, the authors use an analytic approach based on their definitions. Our aim in this note is to provide simpler proofs for these results on spherical convex sets; our proofs are based on some characterizations/representations of spherical convex sets by usual convex sets in Rⁿ. Moreover, we provide a new proof of the recent result of Han and Nashimura on the separation of spherical convex sets established in arXiv:2002.06558. Our proof is based on a result stated in locally convex spaces.

Détail

Classification : 52A55, 52A20, 52A35
Mots-clés : Spherical convexity, Helly theorem, Caratheodory theorem, separation theorem