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 up to 1. All possible convex combinations will be within the convex hull of the given points. In fact, the collection of all such convex combinations of points in the set constitutes the set's 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 satisfy and
As a particular example, every convex combination of two points lies on the line segment between the points.
There exists subsets of a vector space that are not closed under linear combinations but that 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).