Convex combination

Short description: Linear combination of points where all coefficients are non-negative and sum to 1

Given three points [math]\displaystyle{ x_1, x_2, x_3 }[/math] in a plane as shown in the figure, the point [math]\displaystyle{ P }[/math] is a convex combination of the three points, while [math]\displaystyle{ Q }[/math] is not.

([math]\displaystyle{ Q }[/math] is however an affine combination of the three points, as their affine hull is the entire plane.)

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

Convex combination of three points [math]\displaystyle{ v_{0},v_{1},v_{2} \text{ of } 2\text{-simplex} \in \mathbb{R}^{2} }[/math] in a two dimensional vector space [math]\displaystyle{ \mathbb{R}^{2} }[/math] as shown in animation with [math]\displaystyle{ \alpha^{0}+\alpha^{1}+\alpha^{2}=1 }[/math], [math]\displaystyle{ P( \alpha^{0},\alpha^{1},\alpha^{2} ) }[/math] [math]\displaystyle{ := \alpha^{0} v_{0} + \alpha^{1} v_{1} + \alpha^{2} v_{2} }[/math] . When P is inside of the triangle [math]\displaystyle{ \alpha_{i}\ge 0 }[/math]. Otherwise, when P is outside of the triangle, at least one of the [math]\displaystyle{ \alpha_{i} }[/math] is negative.

Convex combination of four points [math]\displaystyle{ A_{1},A_{2},A_{3},A_{4} \in \mathbb{R}^{3} }[/math] in a three dimensional vector space [math]\displaystyle{ \mathbb{R}^{3} }[/math] as animation in Geogebra with [math]\displaystyle{ \sum_{i=1}^{4} \alpha_{i}=1 }[/math] and [math]\displaystyle{ \sum_{i=1}^{4} \alpha_{i}\cdot A_{i}=P }[/math]. When P is inside of the tetrahedron [math]\displaystyle{ \alpha_{i}\gt =0 }[/math]. Otherwise, when P is outside of the tetrahedron, at least one of the [math]\displaystyle{ \alpha_{i} }[/math] is negative.

Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with [math]\displaystyle{ [a,b]=[-4,7] }[/math] and as the first function [math]\displaystyle{ f:[a,b]\to \mathbb{R} }[/math] a polynomial is defined. [math]\displaystyle{ f(x):= \frac{3}{10} \cdot x^2 - 2 }[/math] A trigonometric function [math]\displaystyle{ g:[a,b]\to \mathbb{R} }[/math] was chosen as the second function. [math]\displaystyle{ g(x):= 2 \cdot \cos(x) + 1 }[/math] The figure illustrates the convex combination [math]\displaystyle{ K(t):= (1-t)\cdot f + t \cdot g }[/math] of [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] 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 [math]\displaystyle{ x_1, x_2, \dots, x_n }[/math] in a real vector space, a convex combination of these points is a point of the form

[math]\displaystyle{ \alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n }[/math]

where the real numbers [math]\displaystyle{ \alpha_i }[/math] satisfy [math]\displaystyle{ \alpha_i\ge 0 }[/math] and [math]\displaystyle{ \alpha_1+\alpha_2+\cdots+\alpha_n=1. }[/math]

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 [math]\displaystyle{ [0,1] }[/math] 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 [math]\displaystyle{ X }[/math] is said to have an [math]\displaystyle{ n }[/math]-component finite mixture distribution if its probability density function is a convex combination of [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ x }[/math] is to be used as the reference origin for defining displacement vectors, then [math]\displaystyle{ x }[/math] is a convex combination of [math]\displaystyle{ n }[/math] points [math]\displaystyle{ x_1, x_2, \dots, x_n }[/math] if and only if the zero displacement is a non-trivial conical combination of their [math]\displaystyle{ n }[/math] respective displacement vectors relative to [math]\displaystyle{ x }[/math].
• 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 trianglr - interaktive illustration
• Convex sum/combination with a hexagon - interaktive illustration
• Convex sum/combination with a tetraeder - interaktive illustration

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