Frank Morgan

Karoly Bezdek has proved that a partition of space into convex polyhedra of bounded diameters containing unit balls has average surface area at least 24/√3 ~ 13.86 and conjectures that the minimum is 12√2 ~ 16.97, given by rhombic dodecahedra. Ken Brakke has computed that if one allows arbitrary cells containing unit balls, the rhombic dodecahedra still beat the Kelvin (~17.83) and Weaire-Phelan (~21.15) foams, but they can morph into a “draped Williams cell” (thumbnail at right from nice picture at Brakke) with average area about 16.957, conjectured to be optimal.