Characterizing the D4 lattice as a sphere packing

Question

Suppose I pack spheres in $\mathbb{R}^4$ in such a way that each touches 24 others. (All spheres in my question are assumed to have equal radius and be non-overlapping.) Does this packing necessarily give the $\mathrm{D}_4$ lattice up to translation, rotation and rescaling? Or more precisely: has this been proved?

I think I know the status of a few related conjectures:

1. In 2003 Oleg Musin proved that the kissing number in 4 dimensions is 24: i.e., the maximum number of equal-sized nonoverlapping spheres that can touch another equal-sized sphere in $\mathbb{R}^4$ is 24.

2. It's conjectured that the only way to achieve the kissing number of 24 in 4 dimensions is to have the spheres centered at the vertices (and center) of a regular 24-cell. This conjecture was still open as of 2018.

3. It's conjectured that the $\mathrm{D}_4$ lattice gives the densest packing of spheres in $\mathbb{R}^4$. This conjecture was open as of 2018, and it will probably be big news if it's ever proved.

4. In 1872 Korkin and Zolatarev proved that the $\mathrm{D}_4$ lattice gives the densest lattice packing of spheres in $\mathbb{R}^4$.

5. In 2017 Oleg Musin reported some progress on the 24-cell conjecture, which says that the minimal volume of any cell in the Voronoi decomposition of a packing of $\mathbb{R}^4$ by unit spheres is at least as large as the volume of a regular 24–cell circumscribed around a unit sphere. Musin says this conjecture would imply conjecture 3.

Would settling any of these settle my question? Would settling my question settle any of these? (I can imagine a positive answer to question 2 would imply a positive answer to my question, but I think it would take at least a bit of work.)

Answer 1

As for question 2:

In a recent preprint by David De Laat, Nando Leijenhorst, and Willem De Muinck Keizer (link), they proof optimality and uniqueness for the $D_4$ root system, i.e. they show that the $D_4$ root system is the unique (up to rotations and rescaling) way to achieve the kissing number of 24.

As for whether this implies that if one has a 4D packing in which all balls touch 24 other balls, the lattice should be $D_4$, I think the answer should be yes.

A sketch would be: Assume all balls have radius 1. Then start with one ball in your packing, give it coordinates $(0,0,0,0)$. The 24 balls touching this one are positioned (after rotation) at:

• 8 at $\pm e_i$, for $i \in \{1,2,3,4\}$ and,
• 16 at $(\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2})$.

Simply pick one, for example $(1,0,0,0)$, and observe that there are 9 balls touching it, $(0,0,0,0)$ and $(\frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2})$, which is more then enough to fix a rotation in 4D space. So it can be extended indefinitely.