Convex combination

File:Convex combination illustration.svg

Given three points <math>x_1, x_2, x_3</math> in a plane as shown in the figure, the point <math>P</math> is a convex combination of the three points, while <math>Q</math> is not. Template:Paragraph(<math>Q</math> is however an affine combination of the three points, as their affine hull is the entire plane.)

File:Convex combination 1 ord with geogebra.gif

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

File:ConvexCombination-2D.gif

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

File:ConvexCombination-3D.gif

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

File:Convex combination 1 ord functions with geogebra.gif

Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with <math>[a,b]=[-4,7]</math> and as the first function <math>f:[a,b]\to \mathbb{R}</math> a polynomial is defined. <math>f(x):= \frac{3}{10} \cdot x^2 - 2 </math> A trigonometric function <math>g:[a,b]\to \mathbb{R}</math> was chosen as the second function. <math>g(x):= 2 \cdot \cos(x) + 1</math> The figure illustrates the convex combination <math>K(t):= (1-t)\cdot f + t \cdot g</math> of <math>f</math> and <math>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.<ref name=rock>Template:Citation</ref> 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 <math>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>\alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n</math>

where the real numbers <math>\alpha_i</math> satisfy <math>\alpha_i\ge 0 </math> and <math>\alpha_1+\alpha_2+\cdots+\alpha_n=1.</math><ref name=rock/>

As a particular example, every convex combination of two points lies on the line segment between the points.<ref name=rock/>

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.<ref name=rock/>

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>[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>X</math> is said to have an <math>n</math>-component finite mixture distribution if its probability density function is a convex combination of <math>n</math> so-called component densities.

Related constructions

• A conical combination is a linear combination with nonnegative coefficients. When a point <math>x</math> is to be used as the reference origin for defining displacement vectors, then <math>x</math> is a convex combination of <math>n</math> points <math>x_1, x_2, \dots, x_n</math> if and only if the zero displacement is a non-trivial conical combination of their <math>n</math> respective displacement vectors relative to <math>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 triangle - interactive illustration
• Convex sum/combination with a hexagon - interactive illustration
• Convex sum/combination with a tetraeder - interactive illustration