Please, if anyone can help me with some useful tips to solve this aim:
Let $K^1,...,K^n$ closed convex sets containing the origin of a normed space $E$, and let $c_1,...c_n$ positive real numbers. Prove that, if $x\in E$ cannot be of the form $x=c_1x_1+⋯c_nx_n$ for all $x_i\in K_i, i=1,...,n$, then there is $f\in E^{\ast}$ (the dual of $E$) such that $f(x)>1$ and $f(y)≤1/c_i$ for all $y\in K_{i}$ and i$=1,...,n$.
The book suggested to consider $K=\sum\limits_{i=1}^{n}c_{i}K_{i}$, prove that $K$ is closed and convex and apply the geometric version. BUT, I simply cannot see how $K$ will be closed.
Thanks!
$\sum c_iK_i$ need not be closed. Let $C$ consist of $\{0\}$ and the points $\frac 1 n e^{\frac i n}$ in $\mathbb C$. Let $K$ be the convex hull of $C$. Since $C$ is compact it follows that $K$ is compact and convex containing the origin. Now $e^{1/n}=n\frac 1 n e^{\frac i n} \to 1$and $1 \notin \{cx: c\geq 0,x \in K\}$. [This is because $\Im z>0$ for all $z \in \{cx: c\geq 0,x \in K\}$].