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.
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
[1] As a particular example, every convex combination of two points lies on the line segment between the 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.
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).
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.)
Convex combination of two points
in a two dimensional vector space
as animation in
Geogebra
with
and
Convex combination of three points
in a two dimensional vector space
as shown in animation with
,
. When P is inside of the triangle
. Otherwise, when P is outside of the triangle, at least one of the
is negative.
Convex combination of four points
in a three dimensional vector space
as animation in
Geogebra
with
and
. When P is inside of the tetrahedron
. Otherwise, when P is outside of the tetrahedron, at least one of the
is negative.
Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with
and as the first function
a polynomial is defined.
A trigonometric function
was chosen as the second function.
The figure illustrates the convex combination
of
and
as graph in red color.
![{\displaystyle t\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31a5c18739ff04858eecc8fec2f53912c348e0e5)
![{\displaystyle [a,b]=[-4,7]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/183bacef4a989104893e2c37017d84af54a1a23f)
![{\displaystyle f:[a,b]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b592d102ccd1ba134d401c5b3ea177baaba3ffac)
![{\displaystyle g:[a,b]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1d5103d0e36767a27715bcfc57137119294aad)
