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Convex combination

From Wikipedia, the free encyclopedia

Given three points  x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}}  in a plane as shown in the figure, the point  P {\displaystyle P}  is a convex combination of the three points, while  Q {\displaystyle Q}  is not. ( Q {\displaystyle Q}  is however an affine combination of the three points, as their affine hull is the entire plane.)
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.)

In convex geometry, 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]

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

Related constructions

  • 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

References

  1. ^ a b c d Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683
This page was last edited on 1 October 2020, at 10:46
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