Representing finite convex geometries by relatively convex sets

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.