Embedding convex geometries and a bound on convex dimension

Michael Richter, Luke Rogers

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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 R^N 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), who shows that any 2-dimensional convex geometry may be embedded as circles in R^2, we show that any convex geometry may be embedded as convex polygons in R^2.
Original languageEnglish
Pages (from-to)1059–1063
Number of pages5
JournalDiscrete Mathematics
Volume340
Issue number5
Early online date23 Nov 2016
DOIs
Publication statusPublished - May 2017

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