Theorems of Carathéodory, Helly, and Tverberg Without Dimension

We prove a no-dimensional version of Carath\'edory's theorem: given an $n$-element set $P\subset \mathbb{R}^d$, a point $a \in \mathrm{conv} P$, 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 $\mathrm{conv} Q$ is less then $\mathrm{diam} P/\sqrt {2r}$. 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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