Representing finite convex geometries by relatively convex sets

TL;DR

This paper investigates the structure of convex geometries formed by relatively convex sets in finite-dimensional vector spaces, establishing the n-Carousel Rule and identifying additional properties specific to 2-dimensional cases.

Contribution

It proves that convex geometries of relatively convex sets satisfy the n-Carousel Rule and introduces a new property related to simplex partitions in 2D geometries.

Findings

Convex geometries of relatively convex sets satisfy the n-Carousel Rule.

A new property similar to the simplex partition property is identified for 2D geometries.

Finite sub-geometries also satisfy the n-Carousel Rule.

Abstract

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.