Take a convex and bounded subset $A$ of a normed vector space $V$ so that the closed ball (defined with the norm of V) $B[0, r] \subset A$ for some $r>0$.
We have as well that if $v \in A$, then $-v \in A .$
We want to show that the map $N: V \rightarrow [0, 1)$, $ N(v) = \inf_k \left\{k>0: v/k \in A\right\}$ is another norm on $V$.
EDIT:
I forgot to mention that the vector space is on $\mathbb{C}$.
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I have shown that $N(v)=0$ iff $v=0$. But I don't know how to make progress in proving the two remaining conditions. Namely $N(av)=|a|N(v)$ and $N(v+u) \le N(v)+N(u)$.
I have tried to derive these from the definition of $N$ and infimum, but I'm not making any progress. Thanks for any help.