On the number of defects in optimal quantizers on closed surfaces: the hexagonal torus

TL;DR

This work develops a method to bound the number of non-hexagonal Voronoi cells in energy-minimizing quantizers on closed surfaces by coupling a general energy lower bound with Löschian-number-based upper bounds from the Goldberg-Coxeter construction, linked through a gap lemma. On the hexagonal torus, the authors prove an upper bound of $O(n^{1/4})$ for defects and a related $O(n^{-3/4})$ bound on the Voronoi-area variance, and they conjecture a tighter $O(\log n)$ bound under a Löschian-gap hypothesis. A key technical device is the interpolation lemma, which propagates partial subsequence bounds to all $n$ while controlling the error via gap functions. The paper also proves stability of the defect bound under small perturbations of minimizers and outlines the remaining challenges for transferring the strategy to the sphere, highlighting deep connections between crystallization-like behavior, number-theoretic gaps, and geometric partitions on curved surfaces.

Abstract

We present a strategy for proving an asymptotic upper bound on the number of defects (non-hexagonal Voronoi cells) in the $n$ generator optimal quantizer on a closed surface (i.e., compact 2-manifold without boundary). The program is based upon a general lower bound on the optimal quantization error and related upper bounds for the Löschian numbers $n$ (the norms of the Eisenstein integers) arising from the Goldberg-Coxeter construction. A gap lemma is used to reduce the asymptotics of the number of defects to precisely the asymptotics for the gaps between Löschian numbers. We apply this strategy on the hexagonal torus and prove that the number of defects is at most $O(n^{1/4})$ -- strictly fewer than surfaces with boundary -- and conjecture (based upon the number-theoretic Löschian gap conjecture) that it is in fact $O(\log n)$. Incidentally, the method also yields a related upper bound on the variance of the areas of the Voronoi cells. We show further that the bound on the number of defects holds in a neighborhood of the optimizers. Finally, we remark on the remaining issues for implementation on the 2-sphere.

Figures (6)

• Figure 1: Best configuration for $n=320$ found in a run of MACN.
• Figure 2: Extremal situation near nice $y \in Y_n$ with high degree vertex $w$. By niceness, some point of $Y_n$ must lie in the shaded region (possibly on its boundary). Meanwhile, since $w$ is a vertex of the cell $D_y$, no point of $Y_n$ can lie on the interior of the solid circle (center $w$, radius $R_w = R^*$). This lets us calculate the maximum possible radius $R^*$.
• Figure 3: Two equivalent representations of the hexagonal torus. The name obviously refers to the representation on the left but the rhombus representation on the right (particularly with side length 1) is more convenient for our purposes. Moreover, our results generalize to other parallelograms akin to this rhombus as discussed at the end of the section.
• Figure 4: Visualization of $\mathop{\mathrm{gap}}\nolimits_\mathscr L$. The left endpoint of the horizontal axis represents $1$ and the base of each square is an interval between consecutive Löschian numbers. Thus, the upper edges of the squares (with their left endpoints only) form the graph of $\mathop{\mathrm{gap}}\nolimits_\mathscr L$. The curve is the graph of of the upper bound $2\sqrt 6 n^{1/4} + 3$.
• Figure 5: The Goldberg-Coxeter construction for $(p,q) = (2,1)$ yields the point configuration $Y_{2,1} \subset \mathbb T$ with $n = \abs{Y_{2,1}} = 7$ whose Voronoi tessellation is the well-known tiling of $\mathbb T$ by seven congruent regular hexagons. (In the proof, we scale by $1/s$ so that the rhombus is congruent for all $p,q$.) The area of each of the two large triangles (with solid edges) is clearly $s^2$ times that of each small dashed triangle, hence the number of hexagons is $2s^2 \cdot 3 / 6 = s^2 = p^2 + pq + q^2$ (the total number of small triangles within the rhombus times the number of vertices per triangle divided by six since each hexagon shares the vertices of six triangles).

Theorems & Definitions (33)

• Definition
• Theorem 2.1: Gruber
• proof
• Corollary 2.2
• Remark 2.1
• proof
• Lemma 2.3
• Remark 2.2
• proof
• Lemma 2.4