Editing Convex combination (section)

{{Short description|Linear combination of points where all coefficients are non-negative and sum to 1}}
[[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''. {{paragraph}}(<math>Q</math> is however an affine combination of the three points, as their [[affine hull]] is the entire plane.)]]
[[File:Convex combination 1 ord with geogebra.gif|thumb|Convex combination of two points <math> v_1,v_2 \in \mathbb{R}^2</math> in a two dimensional vector space  <math>\mathbb{R}^2</math> as animation in [[Geogebra]] with <math>t \in [0,1]</math> and <math> K(t) := (1-t)\cdot v_1 + t \cdot v_2</math> ]]

[[File:ConvexCombination-2D.gif|thumb|Convex combination of three points <math>v_{0},v_{1},v_{2} \text{ of } 2\text{-simplex} \in \mathbb{R}^{2}</math> in a two dimensional vector space <math> \mathbb{R}^{2}</math> as shown in animation with <math>\alpha^{0}+\alpha^{1}+\alpha^{2}=1</math>, <math>P( \alpha^{0},\alpha^{1},\alpha^{2} )</math> <math>:= \alpha^{0} v_{0} + \alpha^{1} v_{1} + \alpha^{2} v_{2}</math> . When P is inside of the triangle <math>\alpha_{i}\ge 0</math>.  Otherwise, when P is outside of the triangle, at least one of the <math>\alpha_{i}</math> is negative. ]]

[[File:ConvexCombination-3D.gif|thumb|Convex combination of four points <math>A_{1},A_{2},A_{3},A_{4} \in \mathbb{R}^{3}</math> in a three dimensional vector space <math> \mathbb{R}^{3}</math> as animation in [[Geogebra]] with <math>\sum_{i=1}^{4} \alpha_{i}=1</math> and <small><math>\sum_{i=1}^{4} \alpha_{i}\cdot A_{i}=P</math></small>.
When P is inside of the tetrahedron <math>\alpha_{i}>=0</math>. Otherwise, when P is outside of the tetrahedron, at least one of the <math>\alpha_{i}</math> is negative.]]

[[File:Convex combination 1 ord functions with geogebra.gif|thumb|Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with <math>[a,b]=[-4,7]</math> and as the first function <math>f:[a,b]\to \mathbb{R}</math> a polynomial is defined. <math>f(x):= \frac{3}{10} \cdot x^2 - 2 </math>
A trigonometric function <math>g:[a,b]\to \mathbb{R}</math> was chosen as the second function. <math>g(x):= 2 \cdot \cos(x) + 1</math>
The figure illustrates the convex combination <math>K(t):= (1-t)\cdot f + t \cdot g</math> of <math>f</math> and <math>g</math> as graph in red color.]]
In [[convex geometry]] and [[Vector space|vector algebra]], 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.<ref name=rock>{{citation
 | last = Rockafellar | first = R. Tyrrell
 | mr = 0274683
 | pages = 11–12
 | publisher = Princeton University Press, Princeton, N.J.
 | series = Princeton Mathematical Series
 | title = Convex Analysis
 | volume = 28
 | year = 1970}}</ref> 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.