Convex combination

en.wikipedia.org/wiki/Convex_combination

凸组合

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.

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

{\displaystyle \alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}}

where the real numbers {\displaystyle \alpha _{i}} satisfy {\displaystyle \alpha _{i}\geq 0} and {\displaystyle \alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.}

As a particular example, every convex combination of two points lies on the line segment between the points.

The convex hull of the given 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 {\displaystyle [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).