A convex combination
is a linear combination
(which can be vectors
, or more generally points in an affine space
) where all coefficients
and sum up to 1. All possible convex combinations will be within the convex hull
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
As a particular example, any convex combination of two points will lie on the straight line segment between the points.
- A conical combination is a linear combination with nonnegative coefficients
- 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 sum is explicitly divided from the linear combination.
- 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.