Representation of convex geometries by circles on the plane. (English) Zbl 1409.52001
Summary: Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an \(n\)-dimensional space equipped with a convex hull operator, by the result of K. Kashiwabara et al. [Comput. Geom. 30, No. 2, 129–144 (2005; Zbl 1113.52002)]. Allowing circles rather than points, as was suggested by
G. Czédli [Discrete Math. 330, 61–75 (2014; Zbl 1295.52004)], may presumably reduce the dimension for representation. This paper introduces a property, the Weak \(2 \times 3\)-Carousel rule, which is satisfied by all convex geometries of circles on the plane, and we show that it does not hold in all finite convex geometries. This raises a number of representation problems for convex geometries, which may allow us to better understand the properties of Euclidean space related to its dimension.
MSC:
| 52A01 | Axiomatic and generalized convexity |
Keywords:
closure system; convex geometry; affine convex geometry; convex dimension; Weak \(2 \times 3\)-Carousel ruleReferences:
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