超平面的中心平移定理及其在作图中的应用
A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing
摘要：
受一个开放的问题，从图形绘制，我们研究了几个划分问题的线和超平面安排。我们证明了一个火腿三明治切割定理：给定R^2中的两组n条直线，存在一条直线l，使得在两个直线集中，对于由l界定的两个半平面，有n^{1/2}条直线在该半平面中成对相交，并且这个界限是紧的;一个中心点定理：对于任何n条直线的集合，存在一个点，使得对于包含该点的任何半平面，有（n/3）^{1/2}条直线在该半平面中成对相交。我们推广了这些结果在高维，并获得了一个中心横截定理，一个相同类型的引理，和一个正部分Erdos-Szekeres定理超平面安排。这是通过制定一个推广的中心横截定理，适用于集函数是更一般的措施。回到图的绘制（和在平面上），我们完全解决了激发我们搜索的开放问题：没有一组n个标记的线对所有n-顶点标记的平面图是通用的。作为一个侧记，我们证明了每一组n（未标记）线是所有n-顶点（未标记）平面图的通用。
英文摘要：
Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l such that in both line sets, for both halfplanes delimited by l, there are n^{1/2} lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are (n/3)^{1/2} of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
