# 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 |
---|---|

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 |