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。集合族 $P\subseteq 2^X$ 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 $A\cap B\in P$；且 (3) 若 $A\in P$ 且 $A\neq X$，则存在元素 $\alpha\in X-A$ 使得 $A\cup\{\alpha\}\in P$。作为非空集合族，凸几何具有明确定义的 VC-维数。文献中针对此类结构专门定义了第二个参数——凸维数。在包含序关系下，凸几何亦构成偏序集，且针对这一更大类结构已定义了另外四个维数参数：Dushnik-Miller 维数、布尔维数、局部维数和分数维数。针对这六个维数参数的每一对组合，本文研究是否存在无穷类凸几何使得其中一个参数有界而另一个无界。