# proof that the convex hull of $S$ is open if $S$ is open

Let $S$ be an open set in some topological vector space^{} $V$. For any sequence of positive real numbers $\mathrm{\Lambda}=({\lambda}_{1},\mathrm{\dots},{\lambda}_{n})$ with ${\sum}_{i=1}^{n}{\lambda}_{i}=1$ define

$${S}_{\mathrm{\Lambda}}=\{x\in V\text{such that}x=\sum _{i=1}^{n}{\lambda}_{i}{s}_{i}\text{for}{s}_{i}\in S\}.$$ |

Then since addition and scalar multiplication are both open maps, each ${S}_{\mathrm{\Lambda}}$ is open. Finally, the convex hull^{} is clearly just

$$\bigcup _{\mathrm{\Lambda}}{S}_{\mathrm{\Lambda}},$$ |

which is therefore open.

Title | proof that the convex hull of $S$ is open if $S$ is open |
---|---|

Canonical name | ProofThatTheConvexHullOfSIsOpenIfSIsOpen |

Date of creation | 2013-03-22 14:09:48 |

Last modified on | 2013-03-22 14:09:48 |

Owner | archibal (4430) |

Last modified by | archibal (4430) |

Numerical id | 6 |

Author | archibal (4430) |

Entry type | Proof |

Classification | msc 47L07 |

Classification | msc 46A55 |