Violator spaces vs closure spaces

Violator Spaces were introduced by J. Matousek et al. in 2008 as generalization of Linear Programming problems. Convex geometries were invented by Edelman and Jamison in 1985 as proper combinatorial abstractions of convexity. Convex geometries are defined by anti-exchange closure operators. We investigate an interrelations between violator spaces and closure spaces and show that violator mapping may be defined by a week version of closure operators. Moreover, we prove that violator spaces with an unique basis satisfies the anti-exchange and the Krein-Milman properties.