In convex geometry and vector algebra, 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. In other words, the operation is equivalent to a weighted average, but whose weights are expressed as proportions, instead of freely chosen weights which require dividing by the sum of the weights to find the weighted average.
Formal definition More formally, given a finite number of points in a real vector space or affine space, a convex combination of these points is a point of the form
where the real numbers satisfy and
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.
There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval 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 A random variable is said to have an -component finite mixture distribution if its probability density function is a convex combination of so-called component densities.
Related constructions
A conical combination is a linear combination with nonnegative coefficients. When a point is to be used as the reference origin for defining displacement vectors, then is a convex combination of points if and only if the zero displacement is a non-trivial conical combination of their respective displacement vectors relative to . 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
Affine hull Carathéodory's theorem (convex hull) Simplex Barycentric coordinate system Convex space
References
External links Convex sum/combination with a triangle - interactive illustration Convex sum/combination with a hexagon - interactive illustration Convex sum/combination with a tetraeder - interactive illustration
de:Linearkombination#Konvexkombination