  In a previous work [J. Chem. Phys. 155, 244111 (2021)], we found
counterexamples to the fundamental Hohenberg-Kohn theorem from
density-functional theory in finite-lattice systems represented by graphs.
Here, we demonstrate that this only occurs at very peculiar and rare densities,
those where density sets arising from degenerate ground states, called
degeneracy regions, touch each other or the boundary of the whole density
domain. Degeneracy regions are shown to generally be in the shape of the convex
hull of an algebraic variety, even in the continuum setting. The geometry
arising between density regions and the potentials that create them is analyzed
and explained with examples that, among other shapes, feature the Roman
surface.
