Krein-Milman Spaces

Yulia Kempner, Vadim E. Levit

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)281-286
Number of pages6
JournalElectronic Notes in Discrete Mathematics
StatePublished - Jul 2018


  • Krein-Milman property
  • antimatroid
  • closure space
  • convex geometry
  • convex space
  • extreme point
  • violator space


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