Disclaimer. I'm not an expert geometer, so feel free to fix my language if I use the wrong words...
Let $S$ be a smooth $(n-1)$-dimensional surface in $\mathbb R^n$, with "inside" $A \subseteq \mathbb R^n$ and "outside" $B \subseteq \mathbb R^n$, i.e $A$ and $B$ share a common boundary $S$, and form a partition of $\mathbb R^n$ otherwise. For example, $S$ could be the surface of a sphere, $A \setminus S$ would be the inside, and $B\setminus S$ would be the outside. For each $x \in \mathbb R^n$, let $$ \eta_S(x) := \underset{z \in \mathbb R^n \mid x + z \in S}{\arg\min}\|z\|, $$ Assume the above map is well-defined (i.e single-valued). Note that $d_S(x):= \|\eta_S(x)\|$ is precisely the distance of the point $x$ from the boundary $S$ and $b_S(x) := x + \eta_S(x)$ is the point on the boundary which is closest to $x$. Also note that $\widehat{\eta}_S := \eta_S(x)/d_S(x)$ is the unit outward normal to $S$ at the point $b_S(x)$.
Define the label $l(x)$ of the point $x$ by $$ l(x) = \begin{cases}0,&\mbox{ if }x \in A \setminus S,\\1,&\mbox{ if }x \in B\setminus S,\\\emptyset, &\mbox{ if }x \in S.\end{cases} $$ Given $r \ge 0$, let $\mathbb B^n(x; r)$ be the ball of radius $r$ centered at $x$.
Definition. We say that $S$ satisfies propery $(P_r)$ at the point $x$ if for every $v \in \mathbb B^n(0; r)$, if $|v^T\widehat{\eta}_S(x)| \ge d_S(x)$, then either $l(x + v) \ne (x)$ or $l(x - v) \ne l(x)$.
That is, $S$ satisfies property $(P_r)$ at the point $x$ if (and only if) in the neighborhood $\mathbb B^n(x;r)$ of $x$, the level set $l^{-1}(\{l(x)\}) := \{x' \in \mathbb R^n \mid l(x') = l(x)\}$ is contained in the region $$ S_x(r):= x + \{v \in \mathbb B^n(0;r) \mid |v^T\widehat{\eta}_S(x)| \le d_S(x)\}, $$ a strip of width $2d_S(x)$ trapped between the hyperplane segments $\pm H_x(r) := x + \{v \in \mathbb B^n(0; r) \mid \pm v^T\widehat{\eta}_S(x) = d_S(x)\}$.
Let $r(S) \in [0, \infty]$ be the supremum of values of $r$ for which $S$ satisfies property $(P_r)$ at every point $x \in \mathbb R^m$.
Question. Is there any geometric significance to $r(S)$ ? Is it linked with any classical notion of curvature of $S$?