- Source: Convex combination
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. 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}}
in a real vector 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}}
where the real numbers
α
i
{\displaystyle \alpha _{i}}
satisfy
α
i
≥
0
{\displaystyle \alpha _{i}\geq 0}
and
α
1
+
α
2
+
⋯
+
α
n
=
1.
{\displaystyle \alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.}
As a particular example, every convex combination of two points lies on the line segment between the points.
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.
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]}
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 random variable
X
{\displaystyle X}
is said to have an
n
{\displaystyle n}
-component finite mixture distribution if its probability density function is a convex combination of
n
{\displaystyle n}
so-called component densities.
Related constructions
A conical combination is a linear combination with nonnegative coefficients. When a point
x
{\displaystyle x}
is to be used as the reference origin for defining displacement vectors, then
x
{\displaystyle x}
is a convex combination of
n
{\displaystyle n}
points
x
1
,
x
2
,
…
,
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}
respective displacement vectors relative to
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
Affine hull
Carathéodory's theorem (convex hull)
Simplex
Barycentric coordinate system
Convex space
References
External links
Convex sum/combination with a trianglr - interactive illustration
Convex sum/combination with a hexagon - interactive illustration
Convex sum/combination with a tetraeder - interactive illustration