convex combination

Let $V$ be some vector space over $ℝ$. 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