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


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?

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.

• 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: Interesting smoothing operation. Surely this must have been investigated, but I am not recalling references... Feb 15 '15 at 22:46
• @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