# exposed points are dense in the extreme points

###### Definition.

Let $K\subset {\mathbb{R}}^{n}$ be a closed convex set. A point $p\in K$ is called an exposed
point if there is an $n-1$ dimensional hyperplane^{} whose intersection with $K$ is $p$ alone.

###### Theorem (Strasziewicz).

Let $K\mathrm{\subset}{\mathrm{R}}^{n}$ be a closed convex set. Then the set of exposed points is dense in the set
of extreme points^{}.

For example, let $C(p)$ denote the closed ball in ${\mathbb{R}}^{2}$ of radius 1 and centered at $p.$ Then take
$K$ to be the convex hull^{} of $C(-1,0)$ and $C(1,0)$. The points $(-1,1),$ $(-1,-1),$ $(1,1),$ and $(1,-1)$ are
extreme points, but they are not exposed points.

Title | exposed points are dense in the extreme points |
---|---|

Canonical name | ExposedPointsAreDenseInTheExtremePoints |

Date of creation | 2013-03-22 17:41:04 |

Last modified on | 2013-03-22 17:41:04 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 4 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 52A99 |

Synonym | Strasziewicz theorem |

Related topic | ExtremePoint |

Defines | exposed point |