# Theorems of Caratheodory, Helly, and Tverberg without dimension

• Imre Bárány (UCL and Rényi Institute)
• Thursday 11 October 2018, 15:00-16:00
• MR12.

Caratheodory’s classic result says that if a point $p$ lies in the convex hull of a set $P \subset R^d$, then it lies in the convex hull of a subset $Q \subset P$ of size at most $d+1$. What happens if we want a subset $Q$ of size $k < d+1$ such that $p \in conv Q$? In general, this is impossible as $conv Q$ is too low dimensional. We offer some remedy: $p$ is close to $conv Q$ for some subset $Q$ of size $k$, in an appropriate sense. Similar results hold for the classic Helly and Tverberg theorems as well. This is joint work with Karim Adiprasito, Nabil Mustafa, and Tamas Terpai.

This talk is part of the Combinatorics Seminar series.