  Let $X$ be a finite set. A family $P$ of subsets of $X$ is called a convex
geometry with ground set $X$ if (1) $\emptyset, X\in P$; (2) $A\cap B\in P$
whenever $A,B\in P$; and (3) if $A\in P$ and $A\neq X$, there is an element
$\alpha\in X-A$ such that $A\cup\{\alpha\}\in P$. As a non-empty family of
sets, a convex geometry has a well defined VC-dimension. In the literature, a
second parameter, called convex dimension, has been defined expressly for these
structures. Partially ordered by inclusion, a convex geometry is also a poset,
and four additional dimension parameters have been defined for this larger
class, called Dushnik-Miller dimension, Boolean dimension, local dimension, and
fractional dimension, espectively.
  For each pair of these six dimension parameters, we investigate whether there
is an infinite class of convex geometries on which one parameter is bounded and
the other is not.
