$D_r(x)$ 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
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