Convex combination

Convex combination of three points $v_{0},v_{1},v_{2}{\text{ of }}2{\text{-simplex}}\in \mathbb {R} ^{2}$ in a two dimensional vector space $\mathbb {R} ^{2}$ 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 $\alpha _{i}\geq 0$ . Otherwise, when P is outside of the triangle, at least one of the $\alpha _{i}$ is negative.

Convex combination of four points $A_{1},A_{2},A_{3},A_{4}\in \mathbb {R} ^{3}$ in a three dimensional vector space $\mathbb {R} ^{3}$ 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.

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.

Formal definition

More formally, given a finite number of points $x_{1},x_{2},\dots ,x_{n}$ in a real vector 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 $\alpha _{i}\geq 0$ and $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.$ ^[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 $[0,1]$ 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.

References

^ ^a ^b ^c ^d Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683

External links

Convex sum/combination with a trianglr - interactive illustration
Convex sum/combination with a hexagon - interactive illustration
Convex sum/combination with a tetraeder - interactive illustration

[rock-1] Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683

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