# Order extreme points and solid convex hulls

## TL;DR

This paper extends classical Banach space geometry concepts to an order setting, establishing an order Krein-Milman theorem and exploring properties of order extreme points and solid convex hulls.

## Contribution

It introduces order analogues of extreme points and convex hulls, proves an order Krein-Milman theorem, and links the solid Krein-Milman property to the Radon-Nikodym Property.

## Key findings

- Unit ball of infinite dimensional reflexive space has uncountably many order extreme points.
- Established an order Krein-Milman theorem using a Hahn-Banach type separation.
- Solid Krein-Milman property is equivalent to the Radon-Nikodym Property.

## Abstract

We consider the "order" analogues of some classical notions of Banach space geometry: extreme points and convex hulls. A Hahn-Banach type separation result is obtained, which allows us to establish an "order" Krein-Milman Theorem. We show that the unit ball of any infinite dimensional reflexive space contains uncountably many order extreme points, and investigate the set of positive norm-attaining functionals. Finally,we introduce the "solid" version of the Krein-Milman Property, and show it is equivalent to the Radon-Nikodym Property.