# Theorems of Carathéodory, Helly, and Tverberg without dimension

## Imre Bárány, Rényi Institute, Budapest and University College London

## Date and time: 6pm, Tuesday, April 2, 2019

## Place: Courant Institute, WWH1314

We prove a no-dimensional version of Carathédory's theorem: given an
$n$-element set $P\subset \mathbf{R}^d$, a point $p \in \mathop{\text{conv}}(P)$, and
an integer $r\le d$, $r \le n$, there
is a subset $Q \subset P$ of $r$ elements such that the distance between $p$ and
$\mathop{\text{conv}}(Q)$ is less than $\mathop{\text{diam}}(P)/\sqrt{2r}$. We also give effective algorithms
that find such a set $Q$. An analogous no-dimension Helly theorem says
that, given k⩽d and a finite family $F$ of convex bodies, all contained
in the Euclidean unit ball of $\mathbf{R}^d$, such that every $k$-tuple of the in
the family has a point in common, there is a point q∈R^d which is
closer than $1/\sqrt{k}$ to every set in $F$. This result has several
colourful and fractional consequences. Similar versions of Tverberg's
theorem and some of their extensions are also established.

This is joint work with Karim Adiprasito, Nabil H. Mustafa, and Tamás
Terpai.