It is not true in general that $d_\Omega$ is convex, however smooth the boundary is. E.g., if $\Omega$ is the unit ball centered at the origin, then $d_\Omega(x)=1-|x|$ is concave in $x\in\Omega$.

It is not true in general that $d_\Omega$ is convex, however smooth the boundary is. E.g., if $n\ge2$ and $\Omega$ is the unit ball centered at the origin, then $d_\Omega(x)=1-|x|$ is concave in $x\in\Omega$, and it is not convex in $x$ in any neighborhood of the boundary.