Convex combination

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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 illustration.svg
Given three points in a plane as shown in the figure, the point is a convex combination of the three points, while is not.
( is however an affine combination of the three points, as their affine hull is the entire plane.)

In convex geometry, 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]


More formally, given a finite number of points in a real vector space, a convex combination of these points is a point of the form

where the real numbers satisfy and [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 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).

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In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points, it contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersect every line into a single line segment . For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.

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In mathematics, an affine combination of x1, ..., xn is a linear combination

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Given a finite number of vectors in a real vector space, a conical combination, conical sum, or weighted sum of these vectors is a vector of the form

In algebra, a multivariate polynomial

Glossary of Lie groups and Lie algebras Wikipedia glossary

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.


  1. 1 2 3 4 Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR   0274683