Gersho’s crystallization conjecture

$\text{For }d=3\text{ and every sufficiently regular absolutely continuous probability measure }\mu\text{ on }\mathbb{R}^3,\text{ every sequence of asymptotically optimal }n\text{-point quantizers for the quadratic quantization functional becomes, after the local rescaling determined by the point density, asymptotically congruent to the Voronoi tessellation of the body-centered cubic lattice.\text{ Equivalently, the optimal quantizers crystallize locally into the BCC pattern, whose Voronoi cells are truncated octahedra.}$

References

arXiv

Local crystallization in three-dimensional quantization — arXiv — Ata Deniz Aydin, Mikaela Iacobelli

Posted 3 hours ago · edited 2 hours ago

Progress summary

Refreshed 2 hours ago

Open · progress reported

A new preprint reports local stability of the conjectured three-dimensional BCC pattern, but the full conjecture remains open.

Gersho’s conjecture predicts that optimal high-dimensional quantizers become asymptotically congruent copies of a space-filling polytope. The two-dimensional case is known, while the three-dimensional case is conjectured to have BCC geometry.

Known results

Dimension $2$ is proved (Fejes Tóth, Gruber, and others).
In dimension $3$ , BCC is optimal among lattice configurations (Barnes and Sloan).
Numerical evidence supports the three-dimensional BCC conjecture (Du et al., 2005).
A uniform face-count bound reduces the problem to a finite, very large convex computation, without completing it (Choksi and Lu).

October 2026 local BCC stability

Aydin and Iacobelli’s preprint reports positivity of the second variation for a continuum functional describing locally deformed BCC structures, extending stability beyond lattice configurations. This is substantive local evidence, but it does not establish global optimality for arbitrary quantizers; the advance is therefore unverified as a resolution of the conjecture.

Current status (as of October 2026): The two-dimensional case is settled, while the three-dimensional Gersho conjecture remains open; a preprint claims local BCC stability but not a global proof.

Sources

No solutions have been posted yet.