The isoperimetric problem for four-dimensional parallelohedra

Abstract

The honeycomb and Kelvin problems ask how space can be partitioned into cells of equal volume with the least surface area. We resolve this question in four dimensions for convex bodies that tile space by translations, showing that the regular $24$-cell uniquely minimizes surface area at fixed volume. We develop a framework combining an exact deficit identity, valid in every dimension, with a descent argument. As a consequence, the regular $24$-cell also minimizes surface area among such tiles containing a fixed ball.

arXiv comment36 pages. This version streamlines the proof by further improving our machinery from three dimensions. The abridged applications and root-lattice classification will be covered in a project on higher dimensions