# Convex combination

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.[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 ${\displaystyle x_{1},x_{2},\dots ,x_{n}}$  in a real vector space, a convex combination of these points is a point of the form

${\displaystyle \alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}}$

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

As a particular example, every convex combination of two points lies on the line segment between the points.[1]

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.[1]

There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval ${\displaystyle [0,1]}$  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

• A random variable ${\displaystyle X}$  is said to have an ${\displaystyle n}$ -component finite mixture distribution if its probability density function is a convex combination of ${\displaystyle n}$  so-called component densities.
• A conical combination is a linear combination with nonnegative coefficients. When a point ${\displaystyle x}$  is to be used as the reference origin for defining displacement vectors, then ${\displaystyle x}$  is a convex combination of ${\displaystyle n}$  points ${\displaystyle x_{1},x_{2},\dots ,x_{n}}$  if and only if the zero displacement is a non-trivial conical combination of their ${\displaystyle n}$  respective displacement vectors relative to ${\displaystyle x}$ .
• 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 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.