While reading An easy path to convex analysis and applications [PDF], I encountered the following:
Exercise 2.33 Let $$d(x;\Omega) = \inf_{\omega \in \Omega} \| x - \omega \|$$ and $\Omega \subset \mathbb{R}^n$ is a nonempty, closed, convex set. Prove that $$d(x;\Omega) = \sup_{\|v\|\leq 1} \left\{ \langle x, v \rangle - \sigma_\Omega(v) \right\} $$ where $\sigma_\Omega(v) = \sup\limits_{\omega\in\Omega} \langle v, \omega \rangle$ is the support function of $\Omega$.
I write down
$$ \sup_{\|v\|\leq 1}\{ \langle x, v \rangle - \sigma_\Omega(v)\} = \sup_{\|v\|\leq 1}\{\inf_{\omega\in\Omega} \langle x, v \rangle - \langle v, \omega \rangle\} $$
I think if the $\sup$ and $\inf$ switch, it will be obvious, but I can't go further. Any hints on how can I prove this?
Moreover, how can I calculate the subdifferential $\partial d(\bar{x};\Omega) $ at every $\bar{x} \in \mathbb{R}^n$?