Let $E\subset\mathbb R^N$ be non-empty, compact, and convex and let $F : E\to E$ be Lipschitz-continuous with respect to some norm on $\mathbb R^N$. Since all norms on $\mathbb R^N$ are equivalent, $F$ is Lipschitz-continuous with respect to any norm on $\mathbb R^N$. Let $L(N)$ denote the optimal Lipschitz constant with respect to the norm $N$, i.e., $$ L(N) = \sup_{x,y\in E,\,x\neq y}\frac{N(F(x)-F(y))}{N(x-y)}. $$ Moreover, let $L$ be the infimum of all $L(N)$ over the set of norms.
My question is: Does there always exist a norm $N$ such that $L = L(N)$? In other words: is the infimum attained?
What I did: On the set of norms $\cal N$ I defined a metric $$ d(N_1,N_2) := \sup_{x\in E-E}|N_1(x) - N_2(x)|, $$ but it seems that the function $L : (\cal N,d)\to\mathbb R$ is not continuous. Moreover, I don't know whether $(\cal N,d)$ is a compact space. I guess it's not...