Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by
$$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$
with $W$ the LambertW function defined by $W(x e^x)=x$ for all $x\geq 0$.
Define the metric $d(x,y)=\rho(x-y)$ for $x,y\in\mathbb{R}^n$.
Question: Is the metric space $(X,d)$ normable?
My attempt: Since a topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of 0, and since all metric spaces are Haussdorf, it suffices to show that it has a convex bounded neighborhood of 0, which it has, since its unit ball is convex. Is this correct?