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Theorems of Caratheodory and Tverberg with no dimension

Banff International Research Station for Mathematical Innovation and Discovery

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, in an appropriate sense, to $conv Q$ for some subset $Q$ of size $k$. Similar results hold for Tverberg's theorem as well. This is joint work with Nabil Mustafa.

Mathematics; Convex and discrete geometry; Combinatorics

video/mp4

Author affiliation: Alfred Renyi Institute of Mathematics

2018-08-06

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/66675

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/