# extreme point

###### Definition.

Let $C$ be a convex subset of a vector space $X$. A point $x\in C$ is
called an extreme point^{} if it is not an interior point of any line segment^{}
in $C$. That is $x$ is extreme if and only if whenever $x=ty+(1-t)z$, $t\in (0,1)$, $z\ne y$, implies either $y\notin C$ or $z\notin C$.

For example the set $[0,1]\in \mathbb{R}$ is a convex set and $0$ and $1$ are the extreme points.

## References

- 1 H. L. Royden. . Prentice-Hall, Englewood Cliffs, New Jersey, 1988

Title | extreme point |
---|---|

Canonical name | ExtremePoint |

Date of creation | 2013-03-22 14:24:55 |

Last modified on | 2013-03-22 14:24:55 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 52A99 |

Related topic | FaceOfAConvexSet |

Related topic | ExposedPointsAreDenseInTheExtremePoints |