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

Definition. Let $M \subset \mathbb{R}^n$.
 * $D_r (M): = \bigcup\limits_{x \in M} D_r (x)$
 * $Int_r (M): = \{x ~|~ D_r(x) \subset M\}$

**Question1**: Is it true that $M \subset \mathbb{R}^n$ the f the following conditions are equivalent
 *  There is $r> 0$ such that $Int_r (D_r (M)) = M$
 * M is a [smooth submanifold](https://en.wikipedia.org/wiki/Submanifold#Embedded_submanifolds)

**Question2**: 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