Convex combination Totally Explained

Everything about Convex Combination totally explained

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 up to 1. All possible convex combinations will be within the convex hull of the given points. In fact, the set of all convex combinations constitutes the convex hull.
More formally, given a finite number of points

x_1, x_2, dots, x_n,

in a real vector space, a convex combination of these points is a point of the form ť

alpha_1x_1+alpha_2x_2+cdots+alpha_nx_n

where the real numbers

alpha_i,

satisfy

alpha_ige 0

and

alpha_1+alpha_2+cdots+alpha_n=1.

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

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.

Further Information

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