Convex combination

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Given three points x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} {\displaystyle x_{1},x_{2},x_{3}} in a plane as shown in the figure, the point P {\displaystyle P} {\displaystyle P} is a convex combination of the three points, while Q {\displaystyle Q} {\displaystyle Q} is not.
( Q {\displaystyle Q} {\displaystyle 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 ∈ R 2 {\displaystyle v_{1},v_{2}\in \mathbb {R} ^{2}} {\displaystyle v_{1},v_{2}\in \mathbb {R} ^{2}} in a two dimensional vector space R 2 {\displaystyle \mathbb {R} ^{2}} {\displaystyle \mathbb {R} ^{2}} as animation in Geogebra with t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} {\displaystyle t\in [0,1]} and K ( t ) := ( 1 − t ) ⋅ v 1 + t ⋅ v 2 {\displaystyle K(t):=(1-t)\cdot v_{1}+t\cdot v_{2}} {\displaystyle K(t):=(1-t)\cdot v_{1}+t\cdot v_{2}}
Convex combination of three points v 0 , v 1 , v 2  of  2 -simplex ∈ R 2 {\displaystyle v_{0},v_{1},v_{2}{\text{ of }}2{\text{-simplex}}\in \mathbb {R} ^{2}} {\displaystyle v_{0},v_{1},v_{2}{\text{ of }}2{\text{-simplex}}\in \mathbb {R} ^{2}} in a two dimensional vector space R 2 {\displaystyle \mathbb {R} ^{2}} {\displaystyle \mathbb {R} ^{2}} as shown in animation with α 0 + α 1 + α 2 = 1 {\displaystyle \alpha ^{0}+\alpha ^{1}+\alpha ^{2}=1} {\displaystyle \alpha ^{0}+\alpha ^{1}+\alpha ^{2}=1}, P ( α 0 , α 1 , α 2 ) {\displaystyle P(\alpha ^{0},\alpha ^{1},\alpha ^{2})} {\displaystyle P(\alpha ^{0},\alpha ^{1},\alpha ^{2})} := α 0 v 0 + α 1 v 1 + α 2 v 2 {\displaystyle :=\alpha ^{0}v_{0}+\alpha ^{1}v_{1}+\alpha ^{2}v_{2}} {\displaystyle :=\alpha ^{0}v_{0}+\alpha ^{1}v_{1}+\alpha ^{2}v_{2}} . When P is inside of the triangle α i ≥ 0 {\displaystyle \alpha _{i}\geq 0} {\displaystyle \alpha _{i}\geq 0}. Otherwise, when P is outside of the triangle, at least one of the α i {\displaystyle \alpha _{i}} {\displaystyle \alpha _{i}} is negative.
Convex combination of four points A 1 , A 2 , A 3 , A 4 ∈ R 3 {\displaystyle A_{1},A_{2},A_{3},A_{4}\in \mathbb {R} ^{3}} {\displaystyle A_{1},A_{2},A_{3},A_{4}\in \mathbb {R} ^{3}} in a three dimensional vector space R 3 {\displaystyle \mathbb {R} ^{3}} {\displaystyle \mathbb {R} ^{3}} as animation in Geogebra with ∑ i = 1 4 α i = 1 {\displaystyle \sum _{i=1}^{4}\alpha _{i}=1} {\displaystyle \sum _{i=1}^{4}\alpha _{i}=1} and ∑ i = 1 4 α i ⋅ A i = P {\displaystyle \sum _{i=1}^{4}\alpha _{i}\cdot A_{i}=P} {\displaystyle \sum _{i=1}^{4}\alpha _{i}\cdot A_{i}=P}. When P is inside of the tetrahedron α i >= 0 {\displaystyle \alpha _{i}>=0} {\displaystyle \alpha _{i}>=0}. Otherwise, when P is outside of the tetrahedron, at least one of the α i {\displaystyle \alpha _{i}} {\displaystyle \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 ] {\displaystyle [a,b]=[-4,7]} {\displaystyle [a,b]=[-4,7]} and as the first function f : [ a , b ] → R {\displaystyle f:[a,b]\to \mathbb {R} } {\displaystyle f:[a,b]\to \mathbb {R} } a polynomial is defined. f ( x ) := 3 10 ⋅ x 2 − 2 {\displaystyle f(x):={\frac {3}{10}}\cdot x^{2}-2} {\displaystyle f(x):={\frac {3}{10}}\cdot x^{2}-2} A trigonometric function g : [ a , b ] → R {\displaystyle g:[a,b]\to \mathbb {R} } {\displaystyle g:[a,b]\to \mathbb {R} } was chosen as the second function. g ( x ) := 2 ⋅ cos ⁡ ( x ) + 1 {\displaystyle g(x):=2\cdot \cos(x)+1} {\displaystyle g(x):=2\cdot \cos(x)+1} The figure illustrates the convex combination K ( t ) := ( 1 − t ) ⋅ f + t ⋅ g {\displaystyle K(t):=(1-t)\cdot f+t\cdot g} {\displaystyle K(t):=(1-t)\cdot f+t\cdot g} of f {\displaystyle f} {\displaystyle f} and g {\displaystyle g} {\displaystyle 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.

Formal definition

More formally, given a finite number of points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} {\displaystyle x_{1},x_{2},\dots ,x_{n}} in a real vector space or affine space, a convex combination of these points is a point of the form

α 1 x 1 + α 2 x 2 + ⋯ + α n x n {\displaystyle \alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}} {\displaystyle \alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}}

where the real numbers α i {\displaystyle \alpha _{i}} {\displaystyle \alpha _{i}} satisfy α i ≥ 0 {\displaystyle \alpha _{i}\geq 0} {\displaystyle \alpha _{i}\geq 0} and α 1 + α 2 + ⋯ + α n = 1. {\displaystyle \alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.} {\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 [ 0 , 1 ] {\displaystyle [0,1]} {\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).

Other objects

  • A conical combination is a linear combination with nonnegative coefficients. When a point x {\displaystyle x} {\displaystyle x} is to be used as the reference origin for defining displacement vectors, then x {\displaystyle x} {\displaystyle x} is a convex combination of n {\displaystyle n} {\displaystyle n} points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} {\displaystyle x_{1},x_{2},\dots ,x_{n}} if and only if the zero displacement is a non-trivial conical combination of their n {\displaystyle n} {\displaystyle n} respective displacement vectors relative to x {\displaystyle x} {\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.

See also

References

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