convex combination

Let V be some vector spaceMathworldPlanetmath over . Let X be some set of elements of V. Then a convex combinationMathworldPlanetmath of elements from X is a linear combinationMathworldPlanetmath of the form


for some n>0, where each xiX, each λi0 and iλi=1.

Let co(X) be the set of all convex combinations from X. We call co(X) the convex hull, or convex envelope, or convex closure of X. It is a convex set, and is the smallest convex set which contains X. A set X is convex if and only if X=co(X).

Title convex combination
Canonical name ConvexCombination
Date of creation 2013-03-22 11:50:36
Last modified on 2013-03-22 11:50:36
Owner mps (409)
Last modified by mps (409)
Numerical id 14
Author mps (409)
Entry type Definition
Classification msc 52A01
Synonym convex hull
Synonym convex envelope
Synonym convex closure
Related topic ConvexSet
Related topic AffineCombination