24-cell volume bound for Voronoi cells of arbitrary ball packings

Prove the 24-cell volume conjecture asserting that every Voronoi cell of an arbitrary packing of congruent balls of radius r in four-dimensional Euclidean space has volume at least 8r^4, with equality attained by the regular 24-cell.

Background

The paper derives a volume lower bound of 8r^4 for a convex lattice tile containing a Euclidean ball of radius r. For lattice Voronoi cells, this recovers the optimal lattice packing density and the Hermite bound in dimension four.

The cited 24-cell volume conjecture asks for the same cellwise volume bound for Voronoi cells arising from arbitrary packings of balls, not merely lattice packings. The paper explicitly states that its main theorem does not resolve this conjecture because Voronoi cells of arbitrary packings need not be parallelohedra.