Convex combination

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Image:Convex combination illustration.svg

Given three points <math>x_1, x_2, x_3</math> in a plane as shown in the figure, the point <math>P</math> is a convex combination of the three points, while <math>Q</math> is not (<math>Q</math> is however an affine combination of the three points).

A convex combination is a linear combination of data points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum up to 1. All possible convex combinations (given the base vectors) will be within the convex hull of the given datapoints. In fact, the set of all convex combinations constitutes the convex hull.

More formally, given some points <math>x_1, x_2, \dots, x_n\,</math> in a real vector space, a convex combination of these points is a point of the form

<math>\alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n</math>

where the real numbers <math>\alpha_i\,</math> satisfy <math>\alpha_i\ge 0 </math> and <math>\alpha_1+\alpha_2+\cdots+\alpha_n=1.</math>

As a particular example, any convex combination of two points will lie on the straight line segment between the points.

[edit] Related constructions

Weighted means are functionally the same as convex combinations, but they use a different notation. The coefficients (weights) in a weighted mean are not required to sum to 1; instead the sum is explicitly divided from the linear combination.
Affine combinations are like convex combinations, but the coefficients are not required to be non-negative. Hence affine combinations are defined in vector spaces over any field.