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convex combination (Definition)

Let $ V$ be some vector space over $ \Bbb{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

$\displaystyle \lambda_1 x_1 + \lambda_2 x_2 + \cdots + \lambda_n x_n$
for some $ n > 0$, where each $ x_i \in X$, each $ \lambda_i \ge 0$ and $ \sum_i \lambda_i = 1$.

Let $ {\rm co}(X)$ be the set of all convex combinations from $ X$. We call $ {\rm 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 = {\rm co}(X)$.



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"convex combination" is owned by mps. [ full author list (2) | owner history (1) ]
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See Also: convex set, affine combination

Other names:  convex hull, convex envelope, convex closure

Attachments:
convex hull of $S$ is open if $S$ is open (Theorem) by drini
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Cross-references: contains, convex set, linear combination, vector space
There are 29 references to this entry.

This is version 9 of convex combination, born on 2001-10-19, modified 2006-08-25.
Object id is 401, canonical name is ConvexCombination.
Accessed 21758 times total.

Classification:
AMS MSC: 52A01 (Convex and discrete geometry :: General convexity :: Axiomatic and generalized convexity)
Pending Errata and Addenda
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infinite dimensional vector spaces by jirka on 2004-06-15 17:05:02
Is this definition correct for infinite dimensional vector spaces as well? I'm worried about the sum. A convex combination seems to be a finite dimensional concept.

The definition of convex hull as the intersection of all convex sets containing X most definately works. Perhaps we need such a definition as an alternate for the above? Also we need a definition of closed convex hull which is the intersection of all closed convex sets containing X. I'm now not sure if this is the same as the closure of the convex hull, need to check this out. I can write an alternate entry for such a definition or perhaps this could be incorporated here ...

Jiri
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cross-reference by antizeus on 2001-10-19 22:48:49
Here's an example where we would want control over the cross references -- I would suppress a reference to the "Combinations" object.
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