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).

## Other objects

- Similarly, a convex combination of probability distributions is a weighted sum (where satisfy the same constraints as above) of its component probability distributions, often called a finite mixture distribution, with probability density function:

## Related constructions

- A conical combination is a linear combination with nonnegative coefficients. When a point is to be used as the reference origin for defining displacement vectors, then is a convex combination of points if and only if the zero displacement is a non-trivial conical combination of their respective displacement vectors relative to .
- 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 count 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.