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.
[![CutLocus][1]][1]
*Left*: The cut locus (red) w.r.t. $x$ on a box. *Right*: Unfolding resulting from cutting the cut locus.
(Figure from [Discrete and Computational Geometry](http://press.princeton.edu/titles/9489.html).)
This result generalizes to $\mathbb{R}^d$ for $d > 3$, unfolding without overlap to dimension $d-1$.