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.

翻译：设 $X$ 为有限集合。子集族 $P\subseteq 2^X$ 称为以 $X$ 为基集的凸几何，若满足：(1) $\emptyset, X\in P$；(2) 对任意 $A,B\in P$ 有 $A\cap B\in P$；(3) 若 $A\in P$ 且 $A\neq X$，则存在元素 $\alpha\in X\setminus A$ 使得 $A\cup\{\alpha\}\in P$。作为非空集族，凸几何具有明确定义的VC维。文献中针对该类结构专门定义了另一参数——凸维。在包含序关系下，凸几何同时也是偏序集，为此更大类别定义了四个额外的维度参数，分别称为Dushnik-Miller维、布尔维、局部维和分数维。针对这六个维度参数的所有配对，我们研究是否存在无限类凸几何，使得其中一个参数有界而另一个无界。