Convex combination
Image:Convex combination illustration.svg|right|thumb|Given three points in a plane as shown in the figure, the point is a convex combination of the three points, while is not.
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.
Formal definition
More formally, given a finite number of points in a vector space">vector (geometric)">vector space or affine space, a convex combination of these points is a point of the formwhere 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.
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 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.