# proof of properties of extreme subsets of a closed convex set

For the first claim, it is obvious that ${\bigcap}_{i\in I}{A}_{i}$ is closed convex subset (http://planetmath.org/ConvexSet) of $K$. Let $z\in A$ and $$, $x,y\in K$ such as
$z=tx+(1-t)y$. Then $z\in {A}_{i}$, for all $i\in I$ so we have that $x,y\in {A}_{i}$ for all $i\in I$. Therefore $x,y\in {\bigcap}_{i\in I}{A}_{i}.$

For the second claim suppose $x,y\in K$, $t\in (0,1)$ and $z\in A$ such as $z=tx+(1-t)y$. From the hypothesis $A\subset B$ we have that
$z\in B$ and since $B$ is an extreme subset of $K$, $x,y\in B$. Analogously from the hypothesis that $A$ is an extreme subset of $B$, we have that $x,y\in A$.

Title | proof of properties of extreme subsets of a closed convex set |
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Canonical name | ProofOfPropertiesOfExtremeSubsetsOfAClosedConvexSet |

Date of creation | 2013-03-22 15:25:12 |

Last modified on | 2013-03-22 15:25:12 |

Owner | georgiosl (7242) |

Last modified by | georgiosl (7242) |

Numerical id | 4 |

Author | georgiosl (7242) |

Entry type | Proof |

Classification | msc 52A99 |