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Wikipedia / Convex combination

In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectorss, scalars, or more generally points in an affine space) 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.

Formal definition

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

As a particular example, every convex combination of two points lies on the line segment between the points.

A set is convex if it contains all convex combinations of its points. The convex hull of a given set of points is identical to the set of all their convex combinations.

is convex but generates the real-number line under linear combinations. Another example is the convex set of probability distributions, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one).

Other objects

Related constructions

  • A conical combination is to be used as the reference origin for defining displacement vectors if and only if the zero displacement is a non-trivial conical combination.

  • 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 weighted linear combination is explicitly divided by the sum of the weights.

  • 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.

See also

References

de:Linearkombination#Konvexkombination