Representing finite convex geometries by relatively convex sets

Details Representing finite convex geometries by relatively convex sets Representing finite convex geometries by relatively convex sets

2011-01-07

arxiv.org/abs/1101.1539v1

Mathematics

Combinatorics

12 pages

Scientific paper

A closure system with the anti-exchange axiom is called a convex geometry. One geometry is called a sub-geometry of the other if its closed sets form a sublattice in the lattice of closed sets of the other. We prove that convex geometries of relatively convex sets in $n$-dimensional vector space and their finite sub-geometries satisfy the $n$-Carousel Rule, which is the strengthening of the $n$-Carath$\acute{e}$odory property. We also find another property, that is similar to the simplex partition property and does not follow from $2$-Carusel Rule, which holds in sub-geometries of $2$-dimensional geometries of relatively convex sets.

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