Theorems of Carathéodory, Helly, and Tverberg Without Dimension

Abstract

We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point $a \in {{\,\mathrm{{\texttt {conv}}}\,}}P$, and an integer $r \le n$, there is a subset $Q\subset P$ of r elements such that the distance between a and ${{\,\mathrm{{\texttt {conv}}}\,}}Q$ is less than ${{\,\mathrm{{\texttt {diam}}}\,}}P/\sqrt{2r}$. In an analoguos no-dimension Helly theorem a finite family $\mathcal {F}$ of convex bodies is given, all of them are contained in the Euclidean unit ball of $\mathbb {R}^d$. If $k\le d$, $|\mathcal {F}|\ge k$, and every k-element subfamily of $\mathcal {F}$ is intersecting, then there is a point $q \in \mathbb {R}^d$ which is closer than $1/\sqrt{k}$ to every set in $\mathcal {F}$. This result has several colourful and fractional consequences. Similar versions of Tverberg’s theorem and some of their extensions are also established.