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Convex combination

 

 
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Convex combination
 
 
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A convex combination is a linear combination
Linear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field , with some generalisations given at the end of the article....
 of points
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
 (which can be vectors, scalars
Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
, or more generally points in an affine space
Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine geometry properties of Euclidean space. It can be thought of informally as a vector space where one has forgotten which point is the origin....
) where all coefficients are non-negative and sum up to 1. All possible convex combinations will be within the convex hull
Convex hull

In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....
 of the given points. In fact, the set of all convex combinations constitutes the convex hull.

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

As a particular example, any convex combination of two points will lie on the straight line segment between the points.

Because convex combinations are more restrictive than linear combinations, they define a weaker (more general) theory, and more objects may satisfy them.






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A convex combination is a linear combination
Linear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field , with some generalisations given at the end of the article....
 of points
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
 (which can be vectors, scalars
Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
, or more generally points in an affine space
Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine geometry properties of Euclidean space. It can be thought of informally as a vector space where one has forgotten which point is the origin....
) where all coefficients are non-negative and sum up to 1. All possible convex combinations will be within the convex hull
Convex hull

In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....
 of the given points. In fact, the set of all convex combinations constitutes the convex hull.

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

As a particular example, any convex combination of two points will lie on the straight line segment between the points.

Because convex combinations are more restrictive than linear combinations, they define a weaker (more general) theory, and more objects may satisfy them. Notably, probability distribution
Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
s are closed under convex combination but not linear combination, as the latter do not preserve non-negativity or total integral 1.

Related constructions

  • A conical combination
    Conical combination

    Given a finite number of vectors in a real vector space, a conical combination or a conical sum of these vectors is a vector of the form...
     is a linear combination with nonnegative coefficients
  • Weighted mean
    Weighted mean

    The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others....
    s are functionally the same as convex combinations, but they use a different notation. The coefficients (weights
    Weight function

    A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a "weight" than others....
    ) in a weighted mean are not required to sum to 1; instead the sum is explicitly divided from the linear combination.
  • Affine combination
    Affine combination

    In mathematics, an affine combination of vectors x1, ..., x'n is vectorcalled the linear combination of x1, ..., x'n , in which the sum of the coefficients is 1, thus:...
    s 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
    Field (mathematics)

    In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
    .


See also

  • Carathéodory's theorem (convex hull)
    Carathéodory's theorem (convex hull)

    In convex geometry Carath?odory's theorem states that if a point x of real numberd lies in the convex hull of a set P, there is a subset P′ of P consisting of d+1 or fewer points such that x lies in the convex hull of P′....
  • convex hull
    Convex hull

    In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....