A sandwich with segment convexity

TL;DR

This paper provides a sufficient condition for pairs of functions to have a convex separator in Cartan--Hadamard manifolds or Birkhoff--Beatley systems, and explores unusual convex hull behaviors.

Contribution

It introduces new conditions for convex separation in non-Euclidean geometric structures and studies convex hull properties in these contexts.

Findings

Established a sufficient condition for convex separation in Cartan--Hadamard manifolds.

Analyzed exotic behaviors of convex hulls in the studied structures.

Extended convex analysis concepts to more general geometric settings.

Abstract

The aim of this note is to give a sufficient condition for pairs of functions to have a convex separator when the underlying structure is a Cartan--Hadamard manifold, or more generally: a reduced Birkhoff--Beatley system. Some exotic behavior of convex hulls are also studied.