Geometry of convex geometries

Abstract

We prove that any convex geometry $\mathcal{A}=(U,\mathcal{C})$ on $n$ points and any ideal $\mathcal{I}=(U^{\prime },{\mathcal{C}}^{\prime })$ 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 ${\mathbb{R}}^{O(d)}$.

21 pages, 2 figures