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Back Linearkombination#Konvexkombination German Κυρτός συνδυασμός Greek Combinación convexa Spanish ترکیب محدب Persian Combinaison convexe French צירוף קמור HE Kombinasi cembung ID Combinazione convessa Italian 凸結合 Japanese 볼록 조합 Korean

Given three points $x_{1},x_{2},x_{3}$ 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 $v_{1},v_{2}\in \mathbb {R} ^{2}$ in a two dimensional vector space $\mathbb {R} ^{2}$ as animation in Geogebra with $t\in [0,1]$ and $K(t):=(1-t)\cdot v_{1}+t\cdot v_{2}$

Convex combination of three points $v_{0},v_{1},v_{2}{\text{ of }}2{\text{-simplex}}\in \mathbb {R} ^{2}$ in a two dimensional vector space $\mathbb {R} ^{2}$ as shown in animation with $\alpha ^{0}+\alpha ^{1}+\alpha ^{2}=1$ , $P(\alpha ^{0},\alpha ^{1},\alpha ^{2})$ $:=\alpha ^{0}v_{0}+\alpha ^{1}v_{1}+\alpha ^{2}v_{2}$ . When P is inside of the triangle $\alpha _{i}\geq 0$ . Otherwise, when P is outside of the triangle, at least one of the $\alpha _{i}$ is negative.

Convex combination of four points $A_{1},A_{2},A_{3},A_{4}\in \mathbb {R} ^{3}$ in a three dimensional vector space $\mathbb {R} ^{3}$ as animation in Geogebra with $\sum _{i=1}^{4}\alpha _{i}=1$ and $\sum _{i=1}^{4}\alpha _{i}\cdot A_{i}=P$ . When P is inside of the tetrahedron $\alpha _{i}>=0$ . Otherwise, when P is outside of the tetrahedron, at least one of the $\alpha _{i}$ 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 $f:[a,b]\to \mathbb {R}$ a polynomial is defined. $f(x):={\frac {3}{10}}\cdot x^{2}-2$ A trigonometric function $g:[a,b]\to \mathbb {R}$ was chosen as the second function. $g(x):=2\cdot \cos(x)+1$ The figure illustrates the convex combination $K(t):=(1-t)\cdot f+t\cdot g$ 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.

^ Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683

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