I would like to find a representation for convex hulls co$(\cdot)$ (see wikipedia for the definition of the convex hull) in normed spaces. Let $A,B\subset X$ be bounded and convex subsets of a normed space $X$, then $$co(A\cup B)=\bigcup_{t\in[0,1]}tA+(1-t)B$$ where $tA=\{ta: a\in A\}$ for $t\in[0,1]$ and where $co(A)=\{\sum_{i=1}^nt_ix_i:t_i\ge0, \sum_{i=1}^nt_i=1\text{ and }x_i\in X\}$. I have convinced myself (intuitively) that this equality holds, but I do not know how to write it formally down.
Convex hull of unions
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functional-analysis
banach-spaces
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0What is your definition of $co(S)$ – 2012-04-14
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0@Norbert : You should know from the question that it is the convex hull of $S$. – 2012-04-14
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0Pick an $x$ in the rhs, show it is in the lhs, then repeat the other way around. Alternatively, pick a point in the rhs and show that it must be in any convex set containing $A \cup B$ (hence it would be the convex hull, by definition). – 2012-04-14
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0@PatrickDaSilva You should know that there are different definitions of convex hull – 2012-04-14
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1@Norbert : aren't they all equivalent? I mean, isn't the intersection of all convex sets containing some subset $S$ the same as the set of all convex combinations of elements of $S$? – 2012-04-14
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0Generally it is the smallest convex set containing the set in question (certainly the easiest to state :-)). – 2012-04-14
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0@copper.hat Under that definition, you have to show that the convex hall always exists by showing that the intersection of a collection of convex sets is convex. – 2012-04-14
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0@AlexBecker: That would be true of any definition (the need to show that the resultant object is indeed convex)? Existence of the object is not an issue from a set theoretic standpoint. – 2012-04-14
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0@copper.hat True. I just personally prefer definitions where it's immediately clear that the defined object exists. – 2012-04-14