Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with smooth boundary and set
$$\phi(x)=\operatorname{dist} (x,\partial \Omega) =\inf_{y\in \partial \Omega} |x-y|$$
for $x\in \overline{\Omega}$.
Is there a subset of $\Omega$ where $\phi$ is differentiable?
I'm thinking a subset like $N_s=\{x \in \Omega \mid \operatorname{dist}(x,\partial \Omega) < s\}$.
(for example, in the unitary ball, $\phi$ is not differentiable in the center but it is in $N_s$ for $s<1$)