Distance function to a set

• $d_\Omega(x)=0$ if and only if $x\in\overline{\Omega}$.
• $d_\Omega$ is $1$-Lipschitz:
  $$|d_\Omega(x)-d_\Omega(y)|\leq \|x-y\|.$$
• If $\Omega$ is convex, then $d_\Omega$ is convex.

Conversely, if $\Omega$ is closed and $d_\Omega$ is convex, then $\Omega$ is convex.

• If $\Omega=\{0\}$, then $d_\Omega(x)=\|x\|$.
• If $\Omega$ is the closed ball with center $x_0$ and radius $r$, then
  $$d_\Omega(x)=\max\{\|x-x_0\|-r,0\}.$$