Let $X$ be a vector space, $\|\cdot\|\_1$ and $\|\cdot\|\_2$ two equivalent norms on $X$. Under what further assumptions can we prove there is an isometry between $(X,\|\cdot\|\_1)$ and $(X,\|\cdot\|\_2)$?

In particular, are $(\mathbb{R}^2,\|\cdot\|\_1)$ and $(\mathbb{R}^2,\|\cdot\|\_2)$ isometric (where the two norms are the maximum norm and the usual euclidean norm respectively)?