  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.
