Editing Convex combination (section)

== Formal definition ==
More formally, given a finite number of points <math>x_1, x_2, \dots, x_n</math> in a [[real vector space]], a convex combination of these points is a point of the form
: <math>\alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n</math>
where the real numbers <math>\alpha_i</math> satisfy <math>\alpha_i\ge 0 </math> and <math>\alpha_1+\alpha_2+\cdots+\alpha_n=1.</math><ref name=rock/>

As a particular example, every convex combination of two points lies on the [[line segment]] between the points.<ref name=rock/>

A set is [[convex set|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.<ref name=rock/>

There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval <math>[0,1]</math> is convex but generates the real-number line under linear combinations. Another example is the convex set of [[probability distribution]]s, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one).