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.

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 ${\displaystyle \alpha _{i}\geq 0}$. 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 ${\displaystyle \alpha _{i}>=0}$. 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.

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.

Related constructions

Further information: Linear combination § Affine, conical, and convex combinations

• 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.

See also

• Affine hull
• Carathéodory's theorem (convex hull)
• Simplex
• Barycentric coordinate system
• Convex space