Embedding convex geometries and a bound on convex dimension

Abstract

The notion of an abstract convex geometry, due to Edelman and Jamison (1984), offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara et al. (2005) introduce the notion of a generalized convex shelling into RN and prove that a convex geometry may always be represented with such a shelling. We provide a new, shorter proof of their result using a representation theorem of Edelman and Jamison (1984) and deduce a different upper bound on the dimension of the shelling. Furthermore, in the spirit of Czédli (2014)[6], who shows that any 2-dimensional convex geometry may be embedded as circles in R2, we show that any convex geometry may be embedded as convex polygons in R2.