Convex Combination

linear-algebra convex-geometry

Definition

Convex Combination

Let ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\in V$, where $V$ is a vector space over $\mathbb{R}$, and let ${\lambda }_1,\dots ,{\lambda }_n\in \mathbb{R}$ satisfy

$${\lambda }_i\ge 0\forall i,\sum _{i=1}^n{\lambda }_i=1.$$

Then the sum

$$\sum _{i=1}^n{\lambda }_i\cdot {\mathbf{v}}_i$$

is called a convex combination of ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_n$.

The non-negativity condition ${\lambda }_i\ge 0$ distinguishes convex combinations from general affine combinations. A convex combination with exactly one non-zero coefficient is called a trivial convex combination.