# convex combination

Let $V$ be some vector space^{} over $\mathbb{R}$. Let $X$ be some set of elements of $V$. Then a convex combination^{} of elements from $X$ is a linear combination^{} of the form

$${\lambda}_{1}{x}_{1}+{\lambda}_{2}{x}_{2}+\mathrm{\cdots}+{\lambda}_{n}{x}_{n}$$ |

for some $n>0$, where each ${x}_{i}\in X$, each ${\lambda}_{i}\ge 0$ and ${\sum}_{i}{\lambda}_{i}=1$.

Let $\mathrm{co}(X)$ be the set of all convex combinations from $X$. We call $\mathrm{co}(X)$ the convex hull, or convex envelope, or convex closure of $X$. It is a convex set, and is the smallest convex set which contains $X$. A set $X$ is convex if and only if $X=\mathrm{co}(X)$.

Title | convex combination |
---|---|

Canonical name | ConvexCombination |

Date of creation | 2013-03-22 11:50:36 |

Last modified on | 2013-03-22 11:50:36 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 14 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 52A01 |

Synonym | convex hull |

Synonym | convex envelope |

Synonym | convex closure |

Related topic | ConvexSet |

Related topic | AffineCombination |