This does not fit your non-geometric requirement, but 
nevertheless "might appeal to a group of applied mathematicians":

> If you cut the cut locus of a point $x$ on the surface
of a convex polyhedron $P$ in $\mathbb{R}^3$,
then $P$ unfolds to the plane without self-overlap.

<hr />
[![CutLocus][1]][1]
<br />
<sup>
*Left*: The cut locus (red) w.r.t. $x$ on a box.
*Right*: Unfolding resulting from cutting the cut locus.
</sup>
<hr />

This result generalizes to $\mathbb{R}^d$ for $d > 3$, unfolding
without overlap to dimension $d-1$.


  [1]: https://i.sstatic.net/A9Pv8.jpg