Lopsided sets were introduced by Jim Lawrence in 1983 when he studied the subsets of $\{-1,+1\}^E$ that encode the intersection pattern of a convex set $K$ with the orthants of ${\mathbb R}^E$. Lopsided sets have been independently rediscovered by several other authors, in particular by Andreas Dress in 1995, who called them \emph{ample} sets. Dress defined ample sets as the set families satisfying equality in a combinatorial inequality, which holds for all set families. In a previous article we characterized ample sets in various combinatorial and graph-theoretical ways. In this paper we study geometric realizations of ample sets as cubihedra (cube complexes), which yields several new characterizations. One such characterization establishes that the cubihedra of ample sets endowed with the intrinsic $\ell_1$-metric are exactly the isometric subspaces of $\ell_1$-spaces (which we call, weakly convex sets). We also view the barycenter maps of faces of cubihedra of ample sets as collections of $\{ \pm 1, 0\}$-sign vectors and, in analogy with the characterization of oriented matroids by the covectors and the cocircuits. Moreover, we characterize the collections of $\{ \pm 1, 0\}$-sign vectors corresponding to barycenter maps of all faces and all maximal faces of an ample set. Furthermore, we show that any ample set $\covectors\subseteq \{ -1,+1\}^E$ is realizable as the intersection pattern of a weakly convex set $K$ with the orthants of ${\mathbb R}^E$. All this testifies that the concept of ample sets is quite natural in the context of cube complexes.

翻译：1983年，Jim Lawrence在研究凸集$K$与${\mathbb R}^E$中卦限交集模式时引入了偏集的概念。该概念后被多位学者独立重新发现，特别是Andreas Dress于1995年将其命名为"大集"。Dress将大集定义为满足某种组合不等式等号成立的集族，该不等式对所有集族均成立。在前期工作中，我们从组合与图论角度给出了大集的多重刻画。本文则研究大集在立方体复形中的几何实现，由此得到若干新刻画：其一表明，赋予内蕴$\ell_1$度量的大集立方体复形恰为$\ell_1$空间的等距子空间（我们称之为弱凸集）。通过研究大集立方体复形面的重心映射，我们将其视为$\{ \pm 1, 0\}$-符号向量集合，这与定向拟阵中通过余向量和余回路进行刻画的方式相类似。进一步地，我们给出了对应大集所有面及所有极大面重心映射的$\{ \pm 1, 0\}$-符号向量集合的完整特征。特别地，我们证明任意大集$\covectors\subseteq \{ -1,+1\}^E$均可实现为弱凸集$K$与${\mathbb R}^E$中卦限的交集模式。这些结果充分说明大集概念在立方体复形理论中具有天然地位。