Kelvin problem:

All unit volume, all have polyhedral (and perhaps Voronoi) analogs; all have local analogs.

+ Kelvin best tiling with its combinatorics and all its symmetries?

+ Kelvin best tiling by BCC translations?

+ Kelvin best tiling by translations?

+ Kelvin best isohedral tiling?

(Kelvin best monohedral tiling?)

(Kelvin best equal-pressure?)

+ Kelvin best for Dehn invariant 0 (Hales)?

+ Weaire-Phelan best tiling with its combinatorics and all of its symmetries?

+ Weaire-Phelan best dihedral tiling?

+ Weaire-Phelan best tiling?

+ Kelvin best Voronoi for any lattice (Hales)? Needs bound on area ...

Sullivan: S >= 2/width. Hence if w =< 3/8, mu = (S/2)^3/V^2 ~ 18.963. (WP ~ 18.48).

Lattice must be critical; maybe there are just five, and only Kelvin stable.

+ Can any of these be solved more easily in R^4?