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A convex geometry is a closure system satisfying the anti-exchange property.In this work we document all convex geometries on 4- and 5-element base setswith respect to their representation by circles on the plane. All 34non-isomorphic geometries on a 4-element set can be represented by circles, andof the 672 geometries on a 5-element set, we made representations of 623. Ofthe 49 remaining geometries on a 5-element set, one was already shown not to berepresentable due to the Weak Carousel property, as articulated by Adarichevaand Bolat (Discrete Mathematics, 2019). In this paper we show that 7 more ofthese convex geometries cannot be represented by circles on the plane, due towhat we term the Triangle Property.

Convex geometries representable by at most 5 circles on the plane