Theorems of Carathéodory, Helly, and Tverberg without dimension

Abstract

We prove a no-dimensional version of Carathédory'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.

23 pages, 1 figure