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 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.
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 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 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.