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| {{Unreferenced|date=December 2009}}
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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/>
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| (<math>Q</math> is however an affine combination of the three points, as their [[affine hull]] is the entire plane.)]]
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| 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.
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| 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>
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| As a particular example, every convex combination of two points lies on the [[line segment]] between the points.
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| 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.
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| 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).
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| ==Other objects==
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| *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]]:
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| :<math>f_{X}(x) = \sum_{i=1}^{n} \alpha_i f_{Y_i}(x)</math>
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| ==Related constructions==
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| {{Details|Linear combination#Affine, conical, and convex combinations}}
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| *A [[conical combination]] is a linear combination with nonnegative coefficients
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| *[[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.
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| *[[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]].
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| ==See also==
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| *[[Affine hull]]
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| *[[Carathéodory's theorem (convex hull)]]
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| *[[Convex hull]]
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| *[[Simplex]]
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| *[[Barycentric_coordinate_system_(mathematics)|Barycentric coordinate system]]
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| {{DEFAULTSORT:Convex Combination}}
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| [[Category:Convex geometry]]
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| [[Category:Mathematical analysis]]
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| [[Category:Convex hulls]]
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| [[de:Linearkombination#Spezialfälle]]
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