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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