convex combination

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

 $\lambda_{1}x_{1}+\lambda_{2}x_{2}+\cdots+\lambda_{n}x_{n}$

for some $n>0$, where each $x_{i}\in X$, each $\lambda_{i}\geq 0$ and $\sum_{i}\lambda_{i}=1$.

Let ${\rm co}(X)$ be the set of all convex combinations from $X$. We call ${\rm co}(X)$ 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 $X={\rm co}(X)$.

Title convex combination ConvexCombination 2013-03-22 11:50:36 2013-03-22 11:50:36 mps (409) mps (409) 14 mps (409) Definition msc 52A01 convex hull convex envelope convex closure ConvexSet AffineCombination