Convex combination information

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.

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.

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

\alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}

where the real numbers $\alpha _{i}$ satisfy $\alpha _{i}\geq 0$ and $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.$ ^[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 $[0,1]$ 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 ^b ^c ^d Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683

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

Convex combination information

and 26 Related for: Convex combination information

Convex combination

Linear combination

Convex hull

Convex set

Concave function

Convex polytope

Weighted arithmetic mean

Conical combination

Affine combination

Convex preferences

Locally convex topological vector space

Doubly stochastic matrix

Mixture distribution

Density matrix

Convex cone

Algorithmic problems on convex sets

Convex space

Convex curve

Spokeshave

Integrally convex set

Choquet theory

Quantum depolarizing channel

Separable state

Line segment

Convexity in economics

Arithmetic mean