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 , … , x n {\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
α 1 x 1 + α 2 x 2 + ⋯ + α n x n {\displaystyle \alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}}where the real numbers α i {\displaystyle \alpha _{i}} satisfy α i ≥ 0 {\displaystyle \alpha _{i}\geq 0} and α 1 + α 2 + ⋯ + α n = 1. {\displaystyle \alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.} 2
As a particular example, every convex combination of two points lies on the line segment between the points.3
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.4
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 ] {\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 X {\displaystyle X} is said to have an n {\displaystyle n} -component finite mixture distribution if its probability density function is a convex combination of n {\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 x {\displaystyle x} is to be used as the reference origin for defining displacement vectors, then x {\displaystyle x} is a convex combination of n {\displaystyle n} points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} if and only if the zero displacement is a non-trivial conical combination of their n {\displaystyle n} respective displacement vectors relative to x {\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
Wikiversity has learning resources about Convex combinationExternal 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
References
Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683 /wiki/MR_(identifier) ↩
Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683 /wiki/MR_(identifier) ↩
Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683 /wiki/MR_(identifier) ↩
Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683 /wiki/MR_(identifier) ↩