Gersho’s conjecture in 3D asserts the asymptotic periodicity and structure of the optimal centroidal Voronoi tessellation. This relatively simple crystallization problem remains to date open. We prove bounds on the geometric complexity of optimal centroidal Voronoi tessellations as the number of generators tends to infinity. Combined with an approach of Gruber in 2D, these bounds reduce the resolution of the 3D Gersho’s conjecture to a finite, albeit very large, computation of an explicit convex problem in finitely many variables.

中文翻译：

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3D 中最优质心 Voronoi 细分的几何复杂度的界限

Gersho 的 3D 猜想断言了最佳质心 Voronoi 镶嵌的渐近周期性和结构。这个相对简单的结晶问题至今仍未解决。当生成器的数量趋于无穷大时，我们证明了最优质心 Voronoi 镶嵌的几何复杂性的界限。结合 Gruber 在 2D 中的方法，这些边界将 3D Gersho 猜想的分辨率降低到有限的，尽管非常大，在有限多个变量中的显式凸问题的计算。