Carathéodory’s theorem

Suppose a point $p$ lies in the convex hull of a set $P\subset\mathbb{R}^{d}$. Then there is a subset $P^{\prime}\subset P$ consisting of no more than $d+1$ points such that $p$ lies in the convex hull of $P^{\prime}$.

For example, if a point $p$ is contained in a convex hull of a set $P\subset\mathbb{R}^{2}$, then there are three points in $P$ that determine the triangle containing $p$, provided, of course, that $P$ contains at least three points.

Title Carathéodory’s theorem CaratheodorysTheorem 2013-03-22 13:57:43 2013-03-22 13:57:43 bbukh (348) bbukh (348) 6 bbukh (348) Theorem msc 52A20 ConvexSet