$D_r(x)$ denotes a closed ball of radius $r$ centered at $x$.

Definition. Let $M \subset \mathbb{R}^n$.

• $D_r (M): = \bigcup\limits_{x \in M} D_r (x)$
• $Int_r (M): = \{x ~|~ D_r(x) \subset M\}$

Question: Is it true that $M \subset \mathbb{R}^n$ the following conditions are equivalent

• There is $r> 0$ such that $Int_r (D_r (M)) = M$ and $D_r (Int_r (M)) = M$
• M is a n-dimensional smooth submanifold

Related: https://en.wikipedia.org/wiki/Parallel_curve

Update: This question was previously called question 2 and there was also a trivial question 1 (see edit history), which was refuted in the first comment.