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In convex geometry and vector algebra, a convex combination is a linear combination of points where all coefficients are non-negative and sum to 1. In other words, the operation is equivalent to a standard weighted average, but whose weights are expressed as a percent of the total weight, instead of as a fraction of the count of the weights as in a standard weighted average.
More formally, given a finite number of points x 1 , x 2 , … , x n {\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 α i {\displaystyle \alpha _{i}} satisfy α i ≥ 0 {\displaystyle \alpha _{i}\geq 0} and α 1 + α 2 + ⋯ + α n = 1. {\displaystyle \alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.}
As a particular example, every convex combination of two points lies on the line segment between the points.