Isoperimetric minimization beyond convex translative tiles

Determine the minimum interfacial area among nonconvex fundamental domains and among general equal-volume partitions, extending the four-dimensional regular 24-cell minimization established for parallelohedra.

Background

The paper proves that the regular 24-cell uniquely minimizes surface area among four-dimensional parallelohedra of fixed volume, equivalently among tilings by translates of a single convex tile. It also notes that the regular 24-cell is a candidate for minimizing interfacial area among general equal-volume partitions, while existing results cited in the paper establish only a perturbative form of minimality for general partitions.

The unresolved extension concerns two broader classes excluded from the main theorem: nonconvex fundamental domains and arbitrary equal-volume partitions. Establishing global optimality in either class would substantially extend the four-dimensional isoperimetric theory developed in the paper.