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[[Image:Convex combination illustration.svg|right|thumb|Given three points <math>x_1, x_2, x_3</math> in a plane as shown in the figure, the point <math>P</math> ''is'' a convex combination of the three points, while <math>Q</math> is ''not.''<br/>
(<math>Q</math> 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 [[point (geometry)|points]] (which can be [[vector (geometric)|vector]]s, [[scalar (mathematics)|scalars]], or more generally points in an [[affine space]]) where all [[coefficients]] are [[non-negative]] and sum to 1.


More formally, given a finite number of points <math>x_1, x_2, \dots, x_n\,</math> in a [[real vector space]], a convex combination of these points is a point of the form
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:<math>\alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n</math>
where the real numbers <math>\alpha_i\,</math> satisfy <math>\alpha_i\ge 0 </math> and <math>\alpha_1+\alpha_2+\cdots+\alpha_n=1.</math>
 
As a particular example, every convex combination of two points lies on the [[line segment]] between the points.
 
All convex combinations are within the [[convex hull]] of the given points. In fact, the collection of all such convex combinations of points in the set constitutes the convex hull of the set.
 
There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval <math>[0,1]</math> is convex but generates the real-number line under linear combinations. Another example is the convex set of [[probability distribution]]s, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one).
 
==Other objects==
*Similarly, a convex combination <math>X</math> of [[probability distributions]] <math>Y_i</math> is a weighted sum (where <math>\alpha_i</math> satisfy the same constraints as above) of its component probability distributions, with [[probability density function]]:
 
:<math>f_{X}(x) = \sum_{i=1}^{n} \alpha_i f_{Y_i}(x)</math>
 
==Related constructions==
{{Details|Linear combination#Affine, conical, and convex combinations}}
*A [[conical combination]] is a linear combination with nonnegative coefficients
*[[Weighted mean]]s are functionally the same as convex combinations, but they use a different notation. The coefficients ([[weight function|weights]]) in a weighted mean are not required to sum to 1; instead the sum is explicitly divided from the linear combination.
*[[Affine combination]]s 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 (mathematics)|field]].
 
==See also==
*[[Affine hull]]
*[[Carathéodory's theorem (convex hull)]]
*[[Convex hull]]
*[[Simplex]]
*[[Barycentric_coordinate_system_(mathematics)|Barycentric coordinate system]]
 
{{DEFAULTSORT:Convex Combination}}
[[Category:Convex geometry]]
[[Category:Mathematical analysis]]
[[Category:Convex hulls]]
 
[[de:Linearkombination#Spezialfälle]]

Revision as of 02:53, 1 March 2014

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