Here are the definitions of a radius and center. Let $A\subset G$
$r(A)= \inf _{g \in G} \sup _{g_{1} \in A} ||g- g_{1}||$
If for some $g* \in G$ is $r(A)= \sup _{g_{1} \in A} ||g*- g_{1}||$, then $g*$ is a center of $A$.
So I am looking for a set in a normed space which does not have a center. But most importantly I am wondering how can I tell if a set does not have a center. By contradiction I should show that $r(A)\not= \sup _{g_{1} \in A} ||g*- g_{1}||$. In order to do that I would have to know $r(A)$, but how can I know a radius without knowing the center.
I managed to solve it: For $G$ not complete:
Let $G=\mathbb{Q}$ and $A:=[\sqrt{2}-1,\sqrt{2}+1]\cap\mathbb{Q}$.
Then there is no center in $G$, because if there was, then it would have to be $\sqrt{2}\not\in\mathbb{Q}$