# Convex combination

Short description: Linear combination of points where all coefficients are non-negative and sum to 1
Given three points $\displaystyle{ x_1, x_2, x_3 }$ in a plane as shown in the figure, the point $\displaystyle{ P }$ is a convex combination of the three points, while $\displaystyle{ Q }$ is not.
($\displaystyle{ Q }$ is however an affine combination of the three points, as their affine hull is the entire plane.)
Convex combination of two points $\displaystyle{ v_1,v_2 \in \mathbb{R}^2 }$ in a two dimensional vector space $\displaystyle{ \mathbb{R}^2 }$ as animation in Geogebra with $\displaystyle{ t \in [0,1] }$ and $\displaystyle{ K(t) := (1-t)\cdot v_1 + t \cdot v_2 }$
Convex combination of three points $\displaystyle{ v_{0},v_{1},v_{2} \text{ of } 2\text{-simplex} \in \mathbb{R}^{2} }$ in a two dimensional vector space $\displaystyle{ \mathbb{R}^{2} }$ as shown in animation with $\displaystyle{ \alpha^{0}+\alpha^{1}+\alpha^{2}=1 }$, $\displaystyle{ P( \alpha^{0},\alpha^{1},\alpha^{2} ) }$ $\displaystyle{ := \alpha^{0} v_{0} + \alpha^{1} v_{1} + \alpha^{2} v_{2} }$ . When P is inside of the triangle $\displaystyle{ \alpha_{i}\ge 0 }$. Otherwise, when P is outside of the triangle, at least one of the $\displaystyle{ \alpha_{i} }$ is negative.
Convex combination of four points $\displaystyle{ A_{1},A_{2},A_{3},A_{4} \in \mathbb{R}^{3} }$ in a three dimensional vector space $\displaystyle{ \mathbb{R}^{3} }$ as animation in Geogebra with $\displaystyle{ \sum_{i=1}^{4} \alpha_{i}=1 }$ and $\displaystyle{ \sum_{i=1}^{4} \alpha_{i}\cdot A_{i}=P }$. When P is inside of the tetrahedron $\displaystyle{ \alpha_{i}\gt =0 }$. Otherwise, when P is outside of the tetrahedron, at least one of the $\displaystyle{ \alpha_{i} }$ is negative.
Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with $\displaystyle{ [a,b]=[-4,7] }$ and as the first function $\displaystyle{ f:[a,b]\to \mathbb{R} }$ a polynomial is defined. $\displaystyle{ f(x):= \frac{3}{10} \cdot x^2 - 2 }$ A trigonometric function $\displaystyle{ g:[a,b]\to \mathbb{R} }$ was chosen as the second function. $\displaystyle{ g(x):= 2 \cdot \cos(x) + 1 }$ The figure illustrates the convex combination $\displaystyle{ K(t):= (1-t)\cdot f + t \cdot g }$ of $\displaystyle{ f }$ and $\displaystyle{ g }$ 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.[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 $\displaystyle{ x_1, x_2, \dots, x_n }$ in a real vector space, a convex combination of these points is a point of the form

$\displaystyle{ \alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n }$

where the real numbers $\displaystyle{ \alpha_i }$ satisfy $\displaystyle{ \alpha_i\ge 0 }$ 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.

## Related constructions

• 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.