Prove that $\mathbb{R}^{n}_{1}$ is not isometric to $\mathbb{R}^{n}_{\infty}$ for $n>2$.

Question

$\mathbb{R}^{n}_{1}$ is $\mathbb{R}^{n}$ with norm $\lVert (x_{1},...,x_{n}) \rVert_{1}= |x_{1}|+...+|x_{n}|$.

$\mathbb{R}^{n}_{\infty}$ is $\mathbb{R}^{n}$ with norm $\lVert (x_{1},...,x_{n}) \rVert_{\infty} = \max\{|x_{1}|,...,|x_{n}|\}$.

I know the closed unit ball in the spaces have different shapes when $n>2$. However, I can't use the fact to give a proof.