Let $V$ and $W$ be two normed vector spaces and let $f: V \rightarrow W$ be a map that preserves distance as well as the origin, that is : $ \forall x,y \in V \quad \|f(x)-f(y)\|_W=\|x-y\|_V \quad $ and $\quad f(0_V)=0_W $
Is $f$ necessarily linear ?
I know that if the norm satisfies the parallelogram identity then the answer is yes.
By the Mazur–Ulam theorem, I know that if $f$ is surjective then the answer is also yes. Here I'm asking about the consequences of requiring origin preservation instead of surjectivity.