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. Template:Paragraph($Q$ is however an affine combination of the three points, as their affine hull is the entire plane.)

Convex combination of two points $v_1,v_2\in {ℝ}^2$ in a two dimensional vector space ${ℝ}^2$ as animation in Geogebra with $t\in [0,1]$ and $K(t):=(1-t)\cdot v_1+t\cdot v_2$

Convex combination of three points $v_0,v_1,v_2\text{ of }2\text{-simplex}\in {ℝ}^2$ in a two dimensional vector space ${ℝ}^2$ as shown in animation with ${\alpha }^0+{\alpha }^1+{\alpha }^2=1$, $P({\alpha }^0,{\alpha }^1,{\alpha }^2)$ $:={\alpha }^0{v}_0+{\alpha }^1{v}_1+{\alpha }^2{v}_2$ . When P is inside of the triangle ${\alpha }_i\ge 0$. Otherwise, when P is outside of the triangle, at least one of the ${\alpha }_i$ is negative.

Convex combination of four points $A_1,A_2,A_3,A_4\in {ℝ}^3$ in a three dimensional vector space ${ℝ}^3$ as animation in Geogebra with $\sum _{i=1}^4{\alpha }_i=1$ and $\sum _{i=1}^4{\alpha }_i\cdot A_i=P$. When P is inside of the tetrahedron ${\alpha }_i>=0$. Otherwise, when P is outside of the tetrahedron, at least one of the ${\alpha }_i$ is negative.

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 $f:[a,b]\to ℝ$ a polynomial is defined. $f(x):=\frac{3}{10}\cdot x^2-2$ A trigonometric function $g:[a,b]\to ℝ$ 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.

Formal definition

More formally, given a finite number of points $x_1,x_2,\dots ,x_n$ in a real vector space or affine 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 ${\alpha }_i\ge 0$ 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

• A random variable $X$ is said to have an $n$-component finite mixture distribution if its probability density function is a convex combination of $n$ so-called component densities.

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 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

External links

• Convex sum/combination with a triangle - interactive illustration
• Convex sum/combination with a hexagon - interactive illustration
• Convex sum/combination with a tetraeder - interactive illustration