Convex combination

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Given three points in a plane as shown in the figure, the point is a convex combination of the three points, while is not.
( is however an affine combination of the three points, as their affine hull is the entire plane.)
Convex combination of two points in a two dimensional vector space as animation in Geogebra with and
Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with and as the first function a polynomial is defined. A trigonometric function was chosen as the second function. The figure illustrates the convex combination of and 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.[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 in a real vector space, a convex combination of these points is a point of the form

where the real numbers satisfy and [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 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[edit]

  • Similarly, a convex combination of random variables is a weighted sum (where satisfy the same constraints as above) of its component probability distributions, often called a finite mixture distribution, with probability density function:

Related constructions[edit]

  • A conical combination is a linear combination with nonnegative coefficients. When a point is to be used as the reference origin for defining displacement vectors, then is a convex combination of points if and only if the zero displacement is a non-trivial conical combination of their respective displacement vectors relative to .
  • 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 count 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[edit]


  1. ^ 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