# convex hull of $S$ is open if $S$ is open

Theorem
If $S$ is an open set in a topological vector space^{}, then
the convex hull $\mathrm{co}(S)$ is open.

As the next example shows, the corresponding result does not hold for a closed set^{}.

Example (Valentine, p. 14) If

$$S=\{(x,1/|x|)\in {\mathbb{R}}^{2}\mid x\in \mathbb{R}\setminus \{0\}\},$$ |

then $S$ is closed,
but $\mathrm{co}(S)$ is the open half-space $\{(x,y)\mid x\in \mathbb{R},y\in (0,\mathrm{\infty})\}$,
which is not closed (points on the $x$-axis are accumulation points^{} not in the set, or also can be seen by checking the complement is not open). $\mathrm{\square}$

Reference

F.A. Valentine, *Convex sets*, McGraw-Hill book company, 1964.

Title | convex hull of $S$ is open if $S$ is open |
---|---|

Canonical name | ConvexHullOfSIsOpenIfSIsOpen |

Date of creation | 2013-03-22 13:44:47 |

Last modified on | 2013-03-22 13:44:47 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 9 |

Author | drini (3) |

Entry type | Theorem |

Classification | msc 47L07 |

Classification | msc 46A55 |