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

We prove a no-dimensional version of Carath'edory’s theorem: given an $n$-element set $P\subset {\mathrm{\Re }}^d$, a point $a\in \text{conv}P$, and an integer $r\le d$,$r \le n$,thereisasubset$Q\subset P$of$r$ elements such that the distance between $a$ and $\text{conv}Q$ is less than $\text{diam}P\mathrm{/}\sqrt{2r}$. 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.