Abstract
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.
Original language | English |
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Pages (from-to) | 281-286 |
Number of pages | 6 |
Journal | Electronic Notes in Discrete Mathematics |
Volume | 68 |
DOIs | |
State | Published - Jul 2018 |
Keywords
- Krein-Milman property
- antimatroid
- closure space
- convex geometry
- convex space
- extreme point
- violator space