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 $X=\mathrm{co}(X)$.