论文标题
用轴平行盒实现凸代码
Realizing convex codes with axis-parallel boxes
论文作者
论文摘要
欧几里得空间中每个有序集合的集合都可以与组合代码相关联，该组合记录了空间中的集合所切除的区域。给定两个有序集合的集合，一个可以形成第三个集合，其中$ i $ th集是原始集合中相应集合的笛卡尔产品。我们证明了一个通用的“产品定理”，该代码是根据与原始集合相关的代码来表征与该操作产生的集合相关的代码。我们使用此定理来表征通过轴 - 平行框实现的代码，并在此类代码与凸面开放或封闭集可实现的代码之间表现出差异。我们还使用我们的定理来证明克鲁兹，朱斯蒂，伊斯科夫和克朗霍尔姆的“开放式凸性的单调性”在某些假设略有弱时会固定。
Every ordered collection of sets in Euclidean space can be associated to a combinatorial code, which records the regions cut out by the sets in space. Given two ordered collections of sets, one can form a third collection in which the $i$-th set is the Cartesian product of the corresponding sets from the original collections. We prove a general "product theorem" which characterizes the code associated to the collection resulting from this operation, in terms of the codes associated to the original collections. We use this theorem to characterize the codes realizable by axis-parallel boxes, and exhibit differences between this class of codes and those realizable by convex open or closed sets. We also use our theorem to prove that a "monotonicity of open convexity" result of Cruz, Giusti, Itskov, and Kronholm holds for closed sets when some assumptions are slightly weakened.
