New proofs for some results on spherically convex sets

TL;DR

This paper offers simpler proofs for key results on spherically convex sets by relating them to usual convex sets in Euclidean space, enhancing understanding of spherical convexity.

Contribution

It introduces more straightforward proofs for properties of spherical convex sets using their representations as convex sets in Euclidean space.

Findings

Simpler proofs for properties of spherical convex sets.

Characterizations of spherical convex sets via Euclidean convex sets.

Enhanced understanding of spherical convexity concepts.

Abstract

In Guo and Peng's article [Spherically convex sets and spherically convex functions, J. Convex Anal. 28 (2021), 103--122], one defines the notions of spherical convex sets and functions on "general curved surfaces" in $\mathbb{R}^{n}$ $(n\ge2)$, one studies several properties of these classes of sets and functions, and one establishes analogues of Radon, Helly, Carath\'eodory 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 the results on spherical convex sets; our proofs are based on some characterizations/representations of spherical convex sets by usual convex sets in $\mathbb{R}^{n}$.