  Motivated by recent experiments on bilayer polyhedra composed of amphiphilic
molecules, we study the elastic bending energies of bilayer vesicles forming
polyhedral shapes. Allowing for segregation of excess amphiphiles along the
ridges of polyhedra, we find that bilayer polyhedra can indeed have lower
bending energies than spherical bilayer vesicles. However, our analysis also
implies that, contrary to what has been suggested on the basis of experiments,
the snub dodecahedron, rather than the icosahedron, generally represents the
energetically favorable shape of bilayer polyhedra.
