Convex combination

A weighted average of finitely many points with nonnegative weights summing to one

Convex combination

Let $X$ be a real vector space . A vector $x\in X$ is a convex combination of points $x_1,\dots,x_m\in X$ if there exist scalars $\lambda_1,\dots,\lambda_m\ge 0$ with

\lambda_1+\cdots+\lambda_m=1

such that

x=\sum_{i=1}^m \lambda_i x_i.

Context. Convex combinations describe the points obtained by repeatedly taking “weighted averages.” They generate the convex hull of a set.

Examples:

In $\mathbb{R}^n$ , the point $\tfrac12x_1+\tfrac12x_2$ is the midpoint of $x_1$ and $x_2$ .
If $x_1,x_2,x_3\in\mathbb{R}^2$ are vertices of a triangle, then all points in the triangle are convex combinations of $x_1,x_2,x_3$ .