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Assume that $B$ is an open set, if

\begin{equation*} C:=\{x=\sum_{i=1}^{n}\lambda_{i}x_{i},\lambda_{i} \geq 0,\sum_{i=1}^{n}\lambda_{i}=1,x_i \in B\} \end{equation*}

is a convex that contains $B$, $C$ is an open set?

What's more,$B^c$ denote the convex hull of $B$, $B^c$ is an open set ?

According to the definition of open set, for any $x\in C(B^c)$if we can find $\delta$,s.t. $V(x,\delta)\subset C(B^c)$, then we can prove it. But how to find this $\delta$?

Here is the definition of convex hull http://en.wikipedia.org/wiki/Convex_hull

Thx with any hints!

---------update---------

original: $C:=\{x=\sum_{i=1}^{n}\lambda_{i}x_{i},\lambda_{i} \geq 0,x_i \in B \}$

now: $C:=\{x=\sum_{i=1}^{n}\lambda_{i}x_{i},\lambda_{i} \geq 0,\sum_{i=1}^{n}\lambda_{i}=1, x_i \in B\}$

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    Now, $C$ is the convex hull of $B$ (it's characterization 3. of the definition in the wiki link).2012-12-09

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