Some New Results on Geometric Transversals

IF 0.6 3区 数学 Q4 COMPUTER SCIENCE, THEORY & METHODS Discrete & Computational Geometry Pub Date : 2023-11-16 DOI:10.1007/s00454-023-00573-2
Otfried Cheong, Xavier Goaoc, Andreas F. Holmsen
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引用次数: 1

Abstract

We investigate a number of questions, problems, and conjectures related to geometric transversal theory. Among our results we disprove a conjecture of Bárány and Kalai regarding weak \(\varepsilon \)-nets for k-flats and convex sets in \(\mathbb {R}^d\), and we prove a conjecture of Arocha, Bracho, and Montejano regarding a colorful version of the Goodman–Pollack–Wenger transversal theorem. We also investigate the connected components of the space of line transversals to pairwise disjoint convex sets in \(\mathbb {R}^3\), and we extend a theorem of Karasev and Montejano regarding colorful intersections and k-transversals.

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关于几何截线的一些新结果
我们研究了一些与几何截线理论有关的问题、问题和猜想。在我们的结果中,我们证明了Bárány和Kalai关于\(\mathbb {R}^d\)中k-平坦和凸集的弱\(\varepsilon \) -网的一个猜想,并且证明了Arocha, Bracho和Montejano关于Goodman-Pollack-Wenger截线定理的一个彩色版本的猜想。我们还研究了\(\mathbb {R}^3\)中两两不相交凸集的截线空间的连通分量,并推广了Karasev和Montejano关于彩色交点和k-截线的定理。
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来源期刊
Discrete & Computational Geometry
Discrete & Computational Geometry 数学-计算机:理论方法
CiteScore
1.80
自引率
12.50%
发文量
99
审稿时长
6-12 weeks
期刊介绍: Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.
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