Krein-Milman Spaces

The Krein-Milman theorem characterizes convex subsets in topological vector spaces. Convex geometries were invented as proper combinatorial abstractions of convexity. Further, they turned out to be closure spaces satisfying the Krein-Milman property. Violator spaces were introduced in an attempt to find a general framework for LP-problems. In this work, we investigate interrelations between violator spaces and closure spaces. We prove that a violator space with a unique basis satisfies the Krein-Milman property. Based on subsequent relaxations of the closure operator notion we introduce convex spaces as a generalization of violator spaces and extend the Krein-Milman property to uniquely generated convex spaces.