Is there a term for a polyhedron with $n$ faces (or, similarly, $n$ vertices) that maximises the enclosed volume for a given surface area (equivalently, minimises the surface area for a given volume)?
The equivalent in polygons produces the regular polygons, and can be considered a property of regular polygons. However, as we all know, there are only five "regular polyhedra" (not counting the star polyhedra), the platonic solids. I expect that these five solids maximise for either their respective numbers of faces or vertices (probably not both, as that would result in the dodecahedron and icosahedron having identical volume for the same surface area, which seems counterintuitive), but I'm interested in what the maximising shape would be for, for instance, a pentahedron.
This will wind up similar to the Thomson problem of determining the minimum energy configuration of N electrons on the surface of a sphere.
Those point configurations, which are available from the Wikipedia link, can at least provide a starting point. Put hulls around the points, then calculate surface area and volume. Then perturb the points slightly and recalculate a million times, and see if a better solution pops up. Repeat until perturbing finds a new local minimum. My bet is that all the Thomson configurations will already be local minima. Thomson will be one of these 1. always optimal, 2. always nonoptimal, 3. optimal in certain cases.