Carathéodory’s theorem

Suppose a point $p$ lies in the convex hull of a set $P\subset {ℝ}^d$. 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 $P\subset {ℝ}^2$, 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
Canonical name	CaratheodorysTheorem
Date of creation	2013-03-22 13:57:43
Last modified on	2013-03-22 13:57:43
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	6
Author	bbukh (348)
Entry type	Theorem
Classification	msc 52A20
Related topic	ConvexSet