用轴平行盒实现凸码
Realizing convex codes with axis-parallel boxes
AI总结：
该研究提出组合码的乘积定理，刻画轴平行盒可实现的凸码的特征，对比其与凸开/闭集可实现码的差异，并拓展了开凸性单调性结果的适用范围。
摘要：
欧氏空间中的每个有序集合族都可对应一个组合码，用于记录这些集合在空间中划分出的区域。给定两个有序集合族，可构造第三个集合族，其中第$i$个集合是原两个集合族中对应第$i$个集合的笛卡尔积。我们证明了一个通用的“乘积定理”，该定理用原两个集合族对应的码，刻画了经上述操作得到的集合族所对应的码。我们利用该定理刻画了可由轴平行盒实现的码，并展示了这类码与可由凸开集或凸闭集实现的码之间的差异。我们还利用该定理证明：当略微弱化某些假设时，Cruz、Giusti、Itskov和Kronholm提出的“开凸性单调性”结果对闭集也成立。
英文摘要：
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.
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