Geometry of convex geometries

TL;DR

This paper establishes geometric realizations of convex geometries and related structures as intersections of convex cones with orthants, providing bounds on the complexity and dimension of such realizations.

Contribution

It proves that convex geometries and their ideals can be realized as intersections with convex cones in specific dimensions, linking combinatorial and geometric properties.

Findings

Convex geometries can be realized as intersections with convex cones in R^n.

Realizations are possible with cones having at most m facets, where m relates to critical rooted circuits.

Convex geometries of dimension d are realizable in R^d; multisimplicial complexes in R^{2d}.

Abstract

We prove that any convex geometry $\mathcal{A}=(U,\mathcal{C})$ on $n$ points and any ideal $\mathcal{I}=(U',\mathcal{C}')$ of $\mathcal{A}$ can be realized as the intersection pattern of an open convex polyhedral cone $K\subseteq {\mathbb R}^n$ with the orthants of ${\mathbb R}^n$. Furthermore, we show that $K$ can be chosen to have at most $m$ facets, where $m$ is the number of critical rooted circuits of $\mathcal{A}$. We also show that any convex geometry of convex dimension $d$ is realizable in ${\mathbb R}^d$ and that any multisimplicial complex (a basic example of an ideal of a convex geometry) of dimension $d$ is realizable in ${\mathbb R}^{2d}$ and that this is best possible. From our results it also follows that distributive lattices of dimension $d$ are realizable in ${\mathbb R}^{d}$ and that median systems are realizable. We leave open %the question whether each median system of dimension $d$ is realizable in ${\mathbb R}^{O(d)}$.