关于几何截线的一些新结果

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

摘要

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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Some New Results on Geometric Transversals

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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