Consider a three-dimensional solid such as the Prince Rupert's Cube in this video. This solid has some narrow places where it could easily break.
Now, any solid (even a solid cube) has narrow places (e.g., at the corners of the cube), but we don't consider those breakage risks, and I think the reason is that they're convex. The interesting thing about the narrow "wasp waists" in the 3D-printed artifact above is that they're non-convex. In the language of Maehara & Tokushige (2008), the solid can be "held" by a "frame" around one of these waists.
I'm looking for mathematical terminology to describe either:
What do we call the "waist" points where the area of the smallest "cut" through the solid hits a local minimum?
What do we call the area of that "cut" itself? It strikes me as similar but not the same as diameter in solid geometry. It also strikes me as analogous to minimum cut in graph theory, but I can't really articulate why.
Is there any sense in asking for the "minimum cut" through a nonconvex solid's waists that separates it into two or more pieces? If so, is there a name for that?