Fred Lunnon <fred.lunnon@gmail.com> wrote:
Assuming "randomness" defined in a spherically symmetric manner, the resulting polyhedron will presumably resemble a lumpy sphere with occasional mountains. On the plains in between, most polygons will be triangles at 6-valent vertices. Whether mountains dominate plains would depend on the distribution tail --- I'd guess not for any reasonable extension of normal.
To me, "mountain" implies concavity (at the base of the mountain, not at its peak). So by definition a convex hull would have no mountains. That raises the question of how far above the horizontal a feature could be on such a "planet," where horizontal is defined in reference to the local gravity. And where the local gravity is defined as what you would get if the interior has some fixed density and the exterior has zero density. Which raises the question of efficient ways to calculate the surface gravity on a uniform polyhedron.
Computing the "convex hull" of a point set is a typical example of "geometric sorting" problems, to which computer graphics buffs have devoted considerable attention. Googling (ugh!) those terms should locate plenty of references.
Thanks. I see that key words include "Graham scan," "Chan's algorithm," "Quickhull," "gift wrapping," "monotone chain," and "Jarvis march."