Let $M$ be a submanifold of $\mathbb R^n$, then is there an open set $\Omega$ in $\mathbb R^n$ such that function $d(x,M)$ (distance function) is smooth on $\Omega$?
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Yes. If you choose $\Omega$ to be a small enough normal neighborhood of $M$, then the distance function will be smooth. For a small normal neighborhood, if $x \in \Omega$, then there is a unique vector normal to $M$ that connects $M$ to $x$ and $d(x,M)$ is the length of this vector. Based on this you can show that something called the normal exponential map is a diffeomorphism onto $\Omega$ and this implies that $d$ is smooth in $\Omega$. A complete proof of this fact is proved in many riemannian geometry books (I would check out Bishop and Crittenden's book "Geometry of Manifolds") in the case where $M$ is a hypersurface, and the case where $M$ is not a hypersurface should be similar.