# convex set

Let $S$ a subset of ${\mathbb{R}}^{n}$. We say that $S$ is *convex* when, for any pair of points $A,B$ in $S$, the segment $\overline{AB}$ lies entirely inside $S$.

The former statement is equivalent^{} to saying that for any pair of vectors $u,v$ in $S$, the vector $(1-t)u+tv$ is in $S$ for all $t\in [0,1]$.

If $S$ is a convex set, for any ${u}_{1},{u}_{2},\mathrm{\dots},{u}_{r}$ in $S$, and any positive numbers ${\lambda}_{1},{\lambda}_{2},\mathrm{\dots},{\lambda}_{r}$ such that ${\lambda}_{1}+{\lambda}_{2}+\mathrm{\cdots}+{\lambda}_{r}=1$ the vector

$$\sum _{k=1}^{r}{\lambda}_{k}{u}_{k}$$ |

is in $S$.

Examples of convex sets in the plane are circles, triangles, and ellipses. The definition given above can be generalized to any real vector space:

Let $V$ be a vector space^{} (over $\mathbb{R}$ or $\u2102$). A subset $S$ of $V$
is *convex* if for all points $x,y$ in $S$, the line segment
$\{\alpha x+(1-\alpha )y\mid \alpha \in (0,1)\}$ is also in $S$.

More generally, the same definition works for any vector space over an ordered field.

A *polyconvex set* is a finite union of compact, convex sets.

Remark. The notion of convexity can be generalized to an arbitrary partially ordered set^{}: given a poset $P$ (with partial ordering $\le $), a subset $C$ of $P$ is said to be *convex* if for any $a,b\in C$, if $c\in P$ is between $a$ and $b$, that is, $a\le c\le b$, then $c\in C$.

Title | convex set |

Canonical name | ConvexSet |

Date of creation | 2013-03-22 11:46:35 |

Last modified on | 2013-03-22 11:46:35 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 20 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 52A99 |

Classification | msc 16G10 |

Classification | msc 11F80 |

Classification | msc 22E55 |

Classification | msc 11A67 |

Classification | msc 11F70 |

Classification | msc 06A06 |

Synonym | convex |

Related topic | ConvexCombination |

Related topic | CaratheodorysTheorem2 |

Related topic | ExtremeSubsetOfConvexSet |

Related topic | PropertiesOfExtemeSubsetsOfAClosedConvexSet |

Defines | polyconvex set |

Defines | polyconvex |