Embedding convex geometries and a bound on convex dimension
Abstract
The notion of an abstract convex geometry offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara, Nakamura and Okamoto introduce the notion of a generalized convex shelling into 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 recent representation theorem of Richter and Rubinstein, and deduce a different upper bound on the dimension of the shelling.
Keywords
Cite
@article{arxiv.1502.01941,
title = {Embedding convex geometries and a bound on convex dimension},
author = {Michael Richter and Luke G. Rogers},
journal= {arXiv preprint arXiv:1502.01941},
year = {2016}
}
Comments
- Corrected attribution for Lemma 1 and Theorem 2 - Added an example related to generalized convex shellings of lower-bounded lattices and noted its relevance to convex dimension. - Added a section on embedding convex geometries as convex polygons, including a proof that any convex geometry may be embedded as convex polygons in R^2. - Extended the bibliography. Now 9 pages