Let $\Gamma \subset \mathbb{R}^2$ be a closed simple $C^1$ curve. For every $x \in \mathbb{R}^2\setminus\Gamma$ there exists some $p(x) \in \Gamma$ such that $$ (H) \quad \text{ dist}(x,\Gamma)=|x-p(x)|. $$ Of course $p(x)$ is not necessarily unique. E.g. for $\Gamma=\{x \in \mathbb{R}^2:\, |x|=r\}$ and $x=0$, any $p(0) \in \Gamma$ satisfies (H). However, if $0 < |x| < r$, then $p(x)=rx/|x|$ is the only point satisfying $(H)$.

The question is the following: Is it possible to characterize all closed simple $C^1$ curves $\Gamma \subset \mathbb{R}^2$ with the property:

there is some $\varepsilon_0 > 0$ such that for every $\varepsilon \in (0,\varepsilon_0)$, and every $x \in B_{\varepsilon}(\Gamma)$, there is a unique $p(x) \in \Gamma$ satisfying (H)? where, for $A \subset \mathbb{R}^2$ and $\varepsilon > 0$, $B_{\varepsilon}(A)$ denotes the set $\{x \in \mathbb{R}^2:\, \text{dist}(x,A) < \varepsilon \}$