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 whether each median system of dimension d is realizable in \({{\mathbb {R}}}^{O(d)}\) .
