Geometry of convex geometries

2024/05/21 by Chalopin, Jérémie, Chepoi, Victor, Knauer, Kolja
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.12660

Abstract

We prove that any convex geometry A=(U,C) on n points and any ideal I=(U',C') of A can be realized as the intersection pattern of an open convex polyhedral cone K⊆ \mathbb Rⁿ with the orthants of \mathbb Rⁿ. Furthermore, we show that K can be chosen to have at most m facets, where m is the number of critical rooted circuits of 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).

Related

• Resolutions of Convex Geometries
• Convex geometries representable with colors, by ellipses on the plane, and impossible by circles
• Representation of convex geometries of convex dimension 3 by spheres
• Finite convex geometries of circles
• Representation of convex geometries by circles on a plane