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 system. Some exotic behavior of convex hulls are also studied.
Highlights
As it is well-known, separation theorems play a crucial role in many fields of Analysis and Geometry, and they can be interesting on their own right
Let us quote here the convex separation theorem of Baron, Matkowski, and Nikodem [1], one of our main motivations: Theorem
✩ Supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by the ÚNKP-20-4 New National Excellence Programs of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund, and by the K-134191 NKFIH Grant
Summary
As it is well-known, separation theorems play a crucial role in many fields of Analysis and Geometry, and they can be interesting on their own right. In a recent paper [13], the authors present an extension for functions defined on complete Riemannian manifolds Their generalization is false: As it can be seen, the two-dimensional cases of the main results of [1] and [13] do not coincide. The authors in [13] construct a set as the union of segments joining pairs of points of an epigraph They claim (without explanation) its convexity (page 164, line 7, displayed formula). The original intent of [13] remains a nice and nontrivial challenge: Extend the convex separation theorem of [1] to Riemannian manifolds In this challenge, one has to face two crucial problems. Inequality (1) has to be replaced by another one, in order that an iteration process can be applied
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