A set A with a closure operator c: 2^A → 2^A is called a convex geometry if the empty set is closed and if it satisfies the anti-exchange axiom. This axiom expresses the following property: for every closed set X ⊆ A, if x belongs to the closure of the union of X and a singleton {y}, then y does not belong to the closure of the union of X and the singleton {x}.

The closed subsets of a convex geometry are a lattice. When the closed subsets of a subset B ⊆ A form a sublattice, then B is a subgeometry of A. The particular geometry of a relative convex set is the convex geometry that is constituted on the set of points of the n-dimensional Euclidean space with a particular closure operator.

The main result of the paper shows that convex geometries of relative sets of an n-dimensional Euclidean space and every finite subgeometry satisfy the n-carousel rule, which is a stronger version of the n-Carathéodory property. In the two-dimensional case, it is shown that every convex geometry of relatively convex sets and all its subgeometries satisfy the so-called edge-carousel rule, from which it is concluded that the edge-carousel rule and the two-carousel rule are independent in the two-dimensional case. The final section is devoted to some concluding remarks and an open problem is discussed.

This interesting paper provides a nice contribution to the study of convex geometries.