Embedding convex geometries and a bound on convex dimension

TL;DR

This paper presents a new, concise proof that convex geometries can be represented via generalized convex shellings in real space, offering a tighter upper bound on the shelling dimension.

Contribution

It provides a shorter proof of existing representation results and establishes a new upper bound on the convex dimension of geometries.

Findings

A new, shorter proof of the representation theorem.

A tighter upper bound on the convex dimension.

Application of Richter and Rubinstein's theorem to convex geometries.

Abstract

The notion of an abstract convex geometry offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara, Nakamura and Okamoto introduce the notion of a generalized convex shelling into $\mathbb{R}$ and prove that a convex geometry may always be represented with such a shelling. We provide a new, shorter proof of their result using a recent representation theorem of Richter and Rubinstein, and deduce a different upper bound on the dimension of the shelling.