Let $||\cdot||$ be a norm on $\mathbb{R}^n$, with dual norm $||x||\_\* :=\max\_\limits{y:||y||\leq 1}y^T x$. I'd like to show $$\max\_{x \in \mathbb{R}^n}(x^T d-||x||)=\begin{cases}
0 & \text{ if } ||d||\_\* \leq 1 \\ 
\infty & \text{ otherwise } 
\end{cases}$$
(the Legendre transform of a norm is the characteristic function of the unit ball (wrt the dual norm)   

I'm lost because I don't know how to take the gradient of the objective (and I think $||\cdot||$ is non differentiable anyway). I know that the dual of the dual norm is the original norm: $||\cdot||\_{\*\*} \equiv ||\cdot||$