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