Submitted Paper

Inserted: 2 oct 2026
Last Updated: 2 oct 2026

Year: 2026

Abstract:

We show that the regular $24$-cell uniquely minimizes surface area among four-dimensional parallelohedra of fixed volume. As a consequence, it also minimizes surface area among parallelohedra containing a fixed ball. Moreover, we derive a second-variation formula valid across changes of Voronoi type, which identifies which root-lattice Voronoi cells are strict local minima.