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 up to 1.
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.
All convex combinations are within the convex hull of the given points. In fact, the collection of all such convex combinations of points in the set constitutes the convex hull of the set.
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).
Read more about Convex Combination: Other Objects, Related Constructions
Famous quotes containing the word combination:
“Just as we need to encourage women to test life’s many options, we need to acknowledge real limits of energy and resources. It would be pointless and cruel to prescribe role combination for every woman at each moment of her life. Life has its seasons. There are moments when a woman ought to invest emotionally in many different roles, and other moments when she may need to conserve her psychological energies.”
—Faye J. Crosby (20th century)