Helly numbers of algebraic subsets of ℝd and an extension of Doignon’s Theorem
Helly numbers of algebraic subsets of ℝd and an extension of Doignon’s Theorem
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ℝd 的代数子集的 Helly 数和 Doignon 定理的扩展
DOI：
10.1515/advgeom-2017-0028
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发表时间：
2017
影响因子：
0.5
通讯作者：
E. Roldán
中科院分区：
文献类型：
--
作者：
J. Loera;R. N. L. Haye;Deborah Oliveros;E. Roldán
本文研究了S-凸集，S-凸集是一般凸集与一个真子集S-凸集的交所得到的几何对象，并给出了它们的S-Helly数的新结果。我们扩展了之前关于S = d，d，d−k × k的工作，并给出了几种新情况下的精确界：低维情况，具有某种代数结构的集合，特别是当S是d的任意子群或S是格与其子格之差时。通过抽象Lovász方法的成分，我们得到了许多单色Helly型结果的彩色版本，包括我们自己的结果的几个彩色版本。
Abstract We study S-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in ℝd with a proper subset S ⊂ ℝd, and contribute new results about their S-Helly numbers. We extend prior work for S = ℝd, ℤd, and ℤd−k × ℝk, and give some sharp bounds for several new cases: low-dimensional situations, sets that have some algebraic structure, in particular when S is an arbitrary subgroup of ℝd or when S is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lovász method we obtain colorful versions of many monochromatic Helly-type results, including several colorful versions of our own results.