The Schwartz lantern is a family of sequences of noble polyhedra inscribed within a cylinder that have the counterintuitive property that, as the number of vertices grows larger, the total surface area of the inscribed polyhedra can approach any value (including infinity) that is greater than or equal to the surface area $2\pi r h$ of the circumscribing cylinder.
A cylinder seems to me to be a pretty non-pathological surface, and noble polyhedra seem pretty "nice" and non-pathological as well (as opposed to, say, something weird like the Alexander horned sphere where you could imagine the vertices "bunching up" infinitely faster than average at certain points). If this phenomenon can happen for a surface as simple as a cylinder, then it seems to me that it can probably happen for any regular surface. Is the following conjecture generalizing the Schwarz lantern construction correct?
For any regular surface, there exists a sequence of inscribed polyhedra with an increasing number of vertices whose total surface area approaches any value (including infinity) greater than or equal to the area of the surface.
I also conjecture that as long as the vertices cover every part of the surface with ever-increasing density, then the total surface area of a sequence of inscribed polyhedra can't approach any value less than the surface area of the circumscribing surface. I'm not sure how to make that claim completely rigorous, but I'd be interested if anyone could comment on that conjecture as well.