Test for normability of a metric on a Banach space

Question

If d is a metric on a (finite dimensional) Banach Space and there exist norms $\|-\|_1$ and $\|-\|_2$ and constants $C_1,C_2 \in [1,\infty)$ satisfying:

\begin{equation} C_1\|x-y\|_1 \leq d(x,y) \leq C_2\|x-y\|_2, \end{equation} then is $d$ normable?

By normable I mean there exists some norm $\|\cdot \|$ such that $\|x,y\|=d(x,y)$.

Answer 1

First of all: since all norms on a finite-dimensional space are comparable, the condition could be simpler stated as $$ C_1\|x-y\| \leq d(x,y) \leq C_2\|x-y\| $$ where $\|\cdot \|$ is a norm of our choice, e.g., Euclidean.

Second: the answer is negative, for example $$d(x,y) = |x-y|+\min(|x-y|,1)$$ is a translation-invariant metric on $\mathbb{R}$ that satisfies $$|x-y|\le d(x,y)\le 2|x-y|$$ but is not given by any norm.