Optimal sphere packing in dimension 4

Dimensions 1, 2, 3, 8 and 24 are settled. Dimension 4 is not, and the conjectured answer is the D4 lattice.

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Problem statement

Determine the maximum density of a packing of congruent balls in $\mathbb{R}^4$ . Is the optimum attained by the $D_4$ checkerboard lattice, with density $\pi^2/16 \approx 0.6169$ ?

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Background

Viazovska solved dimension 8 in 2016 using a modular-form construction of a magic function for the Cohn–Elkies linear programming bound; within a week the same team settled dimension 24. Those dimensions are special because $E_8$ and the Leech lattice make the LP bound tight. No comparable structure is available in dimension 4.

Known partial results

The $D_4$ lattice is optimal among lattice packings (Korkine and Zolotareff, 1877).
The Cohn–Elkies LP bound in dimension 4 is close to, but strictly above, the $D_4$ density; numerical work suggests the gap is real, so a different relaxation may be needed.
Cohn–Kumar–Miller–Radchenko–Viazovska proved universal optimality of $E_8$ and Leech, sharpening what "special" means in those dimensions.

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