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arXiv · 1806.08725

Theorems of Carath\'eodory, Helly, and Tverberg without dimension

Karim Adiprasito·Imre Bárány·Nabil H. Mustafa·Tamás Terpai
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Abstract

We prove a no-dimensional version of Carath\'edory's theorem: given an $n$-element set $P\subset \Re^d$, a point $a \in \conv P$, and an integer $r\le d$, $r \le n$, there is a subset $Q\subset P$ of $r$ elements such that the distance between $a$ and $\conv Q$ is less than $\diam P/\sqrt {2r}$. A general no-dimension Helly type result is also proved with colourful and fractional consequences. Similar versions of Tverberg's theorem and some of their extensions are also established.

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Karim Adiprasito, Imre Bárány, Nabil H. Mustafa, Tamás Terpai. 2018-06-22. Theorems of Carath\'eodory, Helly, and Tverberg without dimension. https://arxiv.org/abs/1806.08725

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