Convex combination

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Short description: Linear combination of points where all coefficients are non-negative and sum to 1
Given three points x1,x2,x3 in a plane as shown in the figure, the point P is a convex combination of the three points, while Q is not.
(Q is however an affine combination of the three points, as their affine hull is the entire plane.)
Convex combination of two points v1,v22 in a two dimensional vector space 2 as animation in Geogebra with t[0,1] and K(t):=(1t)v1+tv2
Convex combination of three points v0,v1,v2 of 2-simplex2 in a two dimensional vector space 2 as shown in animation with α0+α1+α2=1, P(α0,α1,α2) :=α0v0+α1v1+α2v2 . When P is inside of the triangle αi0. Otherwise, when P is outside of the triangle, at least one of the αi is negative.
Convex combination of four points A1,A2,A3,A43 in a three dimensional vector space 3 as animation in Geogebra with i=14αi=1 and i=14αiAi=P. When P is inside of the tetrahedron αi>=0. Otherwise, when P is outside of the tetrahedron, at least one of the αi is negative.
Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with [a,b]=[4,7] and as the first function f:[a,b] a polynomial is defined. f(x):=310x22 A trigonometric function g:[a,b] was chosen as the second function. g(x):=2cos(x)+1 The figure illustrates the convex combination K(t):=(1t)f+tg of f and g 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 x1,x2,,xn in a real vector space, a convex combination of these points is a point of the form

α1x1+α2x2++αnxn

where the real numbers αi satisfy αi0 and α1+α2++αn=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.
  • 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 x1,x2,,xn 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.

See also

References

  1. 1.0 1.1 1.2 1.3 Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, N.J., pp. 11–12 

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