# Convex combination 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.) Convex combination of two points $v_{1},v_{2}\in \mathbb {R} ^{2}$ in a two dimensional vector space $\mathbb {R} ^{2}$ as animation in Geogebra with $t\in [0,1]$ and $K(t):=(1-t)\cdot v_{1}+t\cdot v_{2}$  Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with $[a,b]=[4,7]$ and as the first function $f:[a,b]\to \mathbb {R}$ a polynomial is defined. $f(x):={\frac {3}{10}}\cdot x^{2}-2$ A trigonometric function $g:[a,b]\to \mathbb {R}$ was chosen as the second function. $g(x):=2\cdot cos(x)+1$ The figure illustrates the convex combination $K(t):=(1-t)\cdot f+t\cdot g$ of $f$ and $g$ as graph in red color.

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 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.

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.