convex set

Let $S$ a subset of ${ℝ}^n$. 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 $ℝ$ or $ℂ$). 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 $\le$), 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