Convex combination

In convex geometry, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1.

More formally, given a finite number of points ${\displaystyle x_{1},x_{2},\dots ,x_{n}\,}$ in a real vector space, a convex combination of these points is a point of the form

where the real numbers ${\displaystyle \alpha _{i}\,}$ satisfy ${\displaystyle \alpha _{i}\geq 0}$ and ${\displaystyle \alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.}$

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