Gersho’s conjecture in three dimensions

Choose many representative points in a solid cube. Replace each location by its nearest representative and measure the squared distance lost. Does the body-centered cubic pattern—cube corners plus cube centers, repeated through space—give the smallest possible leading average error as the number of representatives grows? This is the three-dimensional question here; it does not claim that every best finite arrangement has exactly that pattern.[1] [3]

Replacing each location in a cube by a nearest representative creates an error that depends on the placement of the representatives. The question is whether the body-centered cubic lattice, with points at repeated cube corners and centers, attains the best possible leading squared-error coefficient among all placements. A simple one-dimensional illustration is the interval [0,1]: representatives at 1/4 and 3/4 give average squared error 1/48, while representatives at 0 and 1 give 1/12. This teaches the objective, not the three-dimensional optimum.

For N representatives Y in the unit cube Q, define $E(Y)=\int_Q\operatorname{dist}(x,Y)^2\,dx$ and $E_N=\inf_{|Y|=N}E(Y)$. Typical cell diameters have scale $N^{-1/3}$, giving squared-error scale $N^{-2/3}$. The conjectured limit of $N^{2/3}E_N$ is the second moment of a unit-volume BCC Voronoi cell about its center: the integral of the full squared distance over that cell. No extra division by three is used. This page isolates that coefficient claim from stronger assertions that every individual optimal cell converges to the same shape.[1] [2] [3]

That pattern is best among lattice arrangements. The open question allows every possible arrangement, including patterns with no repeating lattice. Near the cube’s boundary the cells differ, so the claim concerns the leading error as the number of representatives grows.[1] [3]

The 2025 metric-measure results generalize existence and form of quantization asymptotics without identifying the conjectured three-dimensional cell constant. Choksi and Lu bound the geometric complexity of optimal cells and outline a possible reduction, but do not finish unrestricted optimality. Established BCC lattice optimality compares a restricted class. Boundary cells in a finite cube also prevent a literal statement that every cell is a congruent truncated octahedron at every N.[1] [2] [3]

Replacing continuously varying locations by a finite stored list inevitably loses information. The geometry determines how much average squared error can be avoided for a given supply of representatives. Understanding the best asymptotic coefficient would identify a sharp spatial-approximation limit for uniform volume, rather than merely a good repeating arrangement.[1] [3]

An exact coefficient would turn the dimension-based scaling law into a sharp quantitative limit. The challenge is to control both cell shapes and their ability to fit together: a cell with an attractive individual moment need not tile space compatibly with its neighbors. This coupling explains why local shape optimization and lattice comparisons leave a global geometric problem.[1] [2] [3]