Extreme points in non-positive curvature

TL;DR

This paper investigates the properties of extreme points in non-positively curved spaces, specifically CAT(0) spaces, and demonstrates that a classical theorem analogous to Krein--Milman does not hold in this setting.

Contribution

It provides a counterexample showing that the Krein--Milman theorem's analogue fails in CAT(0) spaces, highlighting limitations in non-positive curvature geometry.

Findings

Krein--Milman analogue fails in CAT(0) spaces

Counterexample demonstrating the failure

Insights into geometric properties of non-positive curvature spaces

Abstract

A natural analogue of the Krein--Milman theorem is shown to fail for CAT(0) spaces.