Intersection Patterns of Convex Sets via Simplicial Complexes: A Survey
Intersection Patterns of Convex Sets via Simplicial Complexes: A Survey
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通过单纯复形的凸集的交集模式：一项调查
DOI：
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发表时间：
2011
期刊：
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通讯作者：
M. Tancer
中科院分区：
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作者：
M. Tancer
本综述的任务是介绍关于凸集交模式的各种结果。研究交叉模式的主要工具之一是通过单纯复形的观点。我们回顾了所谓的d-可表示的，d-可折叠的，和d-勒雷单纯复形，这是非常有用的本研究的定义。我们研究了这些概念之间的差异，也专注于识别它们的计算复杂性。一个列表的海利型定理是在调查。我们还讨论了上述概念对定理所起的重要作用。我们还考虑了良好覆盖的交集模式，它概括了凸集的集合（这些集合可能是“弯曲的”;但是，它们的交集不能太复杂）。我们主要关注新的结果。
The task of this survey is to present various results on intersection patterns of convex sets. One of the main tools for studying intersection patterns is a point of view via simplicial complexes. We recall the definitions of so-called d-representable, d-collapsible, and d-Leray simplicial complexes, which are very useful for this study. We study the differences among these notions and also focus on computational complexity for recognizing them. A list of Helly-type theorems is presented in the survey. We also discuss the important role played by the above-mentioned notions for the theorems. We also consider intersection patterns of good covers, which generalize collections of convex sets (the sets may be “curvy”; however, their intersections cannot be too complicated). We mainly focus on new results.