convex combination
Let V be some vector space over ℝ. Let X be some set of elements of V. Then a convex combination
of elements from X is a linear combination
of the form
λ1x1+λ2x2+⋯+λnxn |
for some n>0, where each xi∈X, each λi≥0 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 |