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BIRS Workshop Lecture Videos

Theorems of Caratheodory and Tverberg with no dimension Bárány, Imre


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.

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