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$\newcommand \X {\mathcal{X}}$ $\newcommand \sd {d_{\rm sign}}$ 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?

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Although your context is broader, these level sets under the Euclidean metric are known as offset polygons. Here is an image from an earlier MO question/answer:

 
Another useful term in this context is the Minkowski sum, e.g., this PDF slide presentation by Andreas Bock: Minkowski Sums and Offsets of Polygons.

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    $\begingroup$ 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? $\endgroup$
    – Travis
    Commented Feb 15, 2015 at 22:14
  • $\begingroup$ @Travis: Interesting smoothing operation. Surely this must have been investigated, but I am not recalling references... $\endgroup$
    – Joseph O'Rourke
    Commented Feb 15, 2015 at 22:46
  • $\begingroup$ @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 $S_{-\sigma}$. $\endgroup$
    – Joseph O'Rourke
    Commented Feb 16, 2015 at 0:04

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