Vtyjkyuk

We consider the normed vector space $(mathbbR^n,VertcdotVert)$.
Is the map $VertcdotVert$ differentiable even if it is not induced by a scalar product?

Sometimes yes, sometimes no. For example, consider the $p$-norms:
$$
|(x_1,dots,x_n)|_p = left(sum_k=1^n |x_k|^pright)^1/p .
$$
This is differentiable (except at the origin) if $p>1$, but not if $p=1$.

A norm on $mathbbR^n$ is never differentiable at the origin. Let $x_0in mathbbR^nsetminus left0right$. Then
$$lim_tto 0fract =lim_tto 0fract|x_0|$$
By the norm axiom $x_0neq 0 Rightarrow |x_0|neq 0$, and so the limit does not exist, as
$$lim_tto 0^+fract=1neq -1=lim_tto 0^-fract $$

On the other hand, the triangle inequality
$$left||x|-|y|right|leq |x-y| $$
tells us that any norm in $mathbbR^n$ is Lipschitz with Lipschitz constant $L=1$. Thus by Rademacher's theorem, it is differentiable almost everywhere in $mathbbR^n$.

A simple example of a norm on $Bbb R^2$ which is non-differentiable at points other than the origin is given by $$||(x,y)||=max(|x|,|y|).$$

If $h>0$ then $||(1+h,1)||=1+h$, while if $-2<h<0$ then $||(1+h,1)||=1$. Hence the function $hmapsto||(1+h,1)||$ is not differentiable at $h=0$, so $||.||$ is not differentiable at $(1,1)$.