Linear combination of points where all coefficients are non-negative and sum to 1
Given three points {\displaystyle x_{1},x_{2},x_{3}} in a plane as shown in the figure, the point {\displaystyle P}is a convex combination of the three points, while {\displaystyle Q} is not. ({\displaystyle Q} is however an affine combination of the three points, as their affine hull is the entire plane.)Convex combination of two points {\displaystyle v_{1},v_{2}\in \mathbb {R} ^{2}} in a two dimensional vector space {\displaystyle \mathbb {R} ^{2}} as animation in Geogebra with {\displaystyle t\in [0,1]} and {\displaystyle K(t):=(1-t)\cdot v_{1}+t\cdot v_{2}}Convex combination of three points {\displaystyle v_{0},v_{1},v_{2}{\text{ of }}2{\text{-simplex}}\in \mathbb {R} ^{2}} in a two dimensional vector space {\displaystyle \mathbb {R} ^{2}} as shown in animation with {\displaystyle \alpha ^{0}+\alpha ^{1}+\alpha ^{2}=1}, {\displaystyle P(\alpha ^{0},\alpha ^{1},\alpha ^{2})}{\displaystyle :=\alpha ^{0}v_{0}+\alpha ^{1}v_{1}+\alpha ^{2}v_{2}} . When P is inside of the triangle {\displaystyle \alpha _{i}\geq 0}. Otherwise, when P is outside of the triangle, at least one of the {\displaystyle \alpha _{i}} is negative. Convex combination of four points {\displaystyle A_{1},A_{2},A_{3},A_{4}\in \mathbb {R} ^{3}} in a three dimensional vector space {\displaystyle \mathbb {R} ^{3}} as animation in Geogebra with {\displaystyle \sum _{i=1}^{4}\alpha _{i}=1} and {\displaystyle \sum _{i=1}^{4}\alpha _{i}\cdot A_{i}=P}. When P is inside of the tetrahedron {\displaystyle \alpha _{i}>=0}. Otherwise, when P is outside of the tetrahedron, at least one of the {\displaystyle \alpha _{i}} is negative.Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with {\displaystyle [a,b]=[-4,7]} and as the first function {\displaystyle f:[a,b]\to \mathbb {R} } a polynomial is defined. {\displaystyle f(x):={\frac {3}{10}}\cdot x^{2}-2} A trigonometric function {\displaystyle g:[a,b]\to \mathbb {R} } was chosen as the second function. {\displaystyle g(x):=2\cdot \cos(x)+1} The figure illustrates the convex combination {\displaystyle K(t):=(1-t)\cdot f+t\cdot g} of {\displaystyle f} and {\displaystyle g} as graph in red color.
More formally, given a finite number of points {\displaystyle x_{1},x_{2},\dots ,x_{n}} in a real vector space, a convex combination of these points is a point of the form
where the real numbers {\displaystyle \alpha _{i}} satisfy {\displaystyle \alpha _{i}\geq 0} and {\displaystyle \alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.}[1]
As a particular example, every convex combination of two points lies on the line segment between the points.[1]
A set is convex if it contains all convex combinations of its points.
The convex hull of a given set of points is identical to the set of all their convex combinations.[1]
There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval {\displaystyle [0,1]} is convex but generates the real-number line under linear combinations. Another example is the convex set of probability distributions, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one).
A random variable {\displaystyle X} is said to have an {\displaystyle n}-component finite mixture distribution if its probability density function is a convex combination of {\displaystyle n} so-called component densities.
A conical combination is a linear combination with nonnegative coefficients. When a point {\displaystyle x} is to be used as the reference origin for defining displacement vectors, then {\displaystyle x} is a convex combination of {\displaystyle n} points {\displaystyle x_{1},x_{2},\dots ,x_{n}} if and only if the zero displacement is a non-trivial conical combination of their {\displaystyle n} respective displacement vectors relative to {\displaystyle x}.
Weighted means are functionally the same as convex combinations, but they use a different notation. The coefficients (weights) in a weighted mean are not required to sum to 1; instead the weighted linear combination is explicitly divided by the sum of the weights.
Affine combinations 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.
