More formally, given a finite number of points $x_{1},x_{2},\dots ,x_{n))$ in a real vector space, a convex combination of these points is a point of the form

where the real numbers $\alpha _{i))$ satisfy $\alpha _{i}\geq 0$ and $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.$^{[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 $[0,1]$ 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).

A conical combination is a linear combination with nonnegative coefficients. When a point $x$ is to be used as the reference origin for defining displacement vectors, then $x$ is a convex combination of $n$ points $x_{1},x_{2},\dots ,x_{n))$ if and only if the zero displacement is a non-trivial conical combination of their $n$ respective displacement vectors relative to $x$.

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.