# convex set

Let $S$ a subset of $\mathbbmss{R}^{n}$. 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 $u_{1},u_{2},\ldots,u_{r}$ in $S$, and any positive numbers $\lambda_{1},\lambda_{2},\ldots,\lambda_{r}$ such that $\lambda_{1}+\lambda_{2}+\cdots+\lambda_{r}=1$ 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 $\mathbbmss{R}$ or $\mathbbmss{C}$). 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 $\leq$), 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, $a\leq c\leq b$, 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