# Convex combination

In convex geometry, 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. Given three points $x_{1},x_{2},x_{3}$ in a plane as shown in the figure, the point $P$ is a convex combination of the three points, while $Q$ is not.
($Q$ is however an affine combination of the three points, as their affine hull is the entire plane.)

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

$\alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}$ where the real numbers $\alpha _{i}$ satisfy $\alpha _{i}\geq 0$ and $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.$ 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 $[0,1]$ 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

• Similarly, a convex combination $X$  of random variables $Y_{i}$  is a weighted sum (where $\alpha _{i}$  satisfy the same constraints as above) of its component probability distributions, often called a finite mixture distribution, with probability density function:
$f_{X}(x)=\sum _{i=1}^{n}\alpha _{i}f_{Y_{i}}(x)$

## Related constructions

• A conical combination is a linear combination with nonnegative coefficients. When a point $x$  is to be used as the reference origin for defining displacement vectors, then $x$  is a convex combination of $n$  points $x_{1},x_{2},\dots ,x_{n}$  if and only if the zero displacement is a non-trivial conical combination of their $n$  respective displacement vectors relative to $x$ .
• 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 count 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.