# Is there a name for the level-sets of the signed distance function to a set in a metric space?

$\newcommand \X {\mathcal{X}}$ $\newcommand \sd {d_{\rm sign}}$ Let $(\X, d)$ be a metric space and define the distance between a point $x \in \X$ and a set $S \subset X$ by $d(x,S) = \inf_{y \in S} d(x,y)$ and the signed distance to be $$\sd(x,S) = \begin{cases} d(x,S) & \hbox{ if x \not \in S}\\ -d(x,S^c) & \hbox{ if x \in S} \end{cases}.$$

For any $\sigma \in \mathbb{R}$, we can define $S_\sigma = \{x \in \X \,:\, \sd(x,S) \leq \sigma\}$. When $\sigma > 0$ this corresponds to enlarging the set $S$ and when $\sigma$ is negative, this corresponds to shrinking $S$. Is there a name for the set $S_\sigma$, and do these sets have well-known properties? • Awesome, thank you. I am also interested in the following operation: Fix some $\sigma > 0$ and map each set $S$ to the double-offset polygon $(S_{-\sigma})_\sigma$ where we first shrink it by a distance $\sigma$ and then enlarge it again. This has the effect of smoothing out any details of the set $S$ that are smaller than $\sigma$. Does this have a name? Feb 15 '15 at 22:14
• @Travis: This is a stretch, but I just remembered that one can smooth a shape using the medial axis: "Shape Smoothing Using Medial Axis Properties," (journal link). And the medial axis is very much related to your $S_{-\sigma}$. Feb 16 '15 at 0:04