  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.
