Representing finite convex geometries by relatively convex sets. (English) Zbl 1282.05022
Summary: 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 \(n\)-Carathéodory property. We also find another property, that is similar to the simplex partition property and independent of 2-carousel rule, which holds in sub-geometries of 2-dimensional geometries of relatively convex sets.
MSC:
| 05B25 | Combinatorial aspects of finite geometries |
| 52A05 | Convex sets without dimension restrictions (aspects of convex geometry) |
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