Embedding convex geometries and a bound on convex dimension

作者：Richter Michael^*; Rogers Luke G

来源：Discrete Mathematics, 2017, 340(5): 1059-1063.

DOI：10.1016/j.disc.2016.10.006

• Convex geometry
• Generalized convex shelling
• Convex dimension
• Convex polygon embedding

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 Czedli (2014)[6], 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.

• 出版日期2017-5