We investigate the Edge-Isoperimetric Problem (EIP) for sets with n elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers M n of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most O ( n 3 / 4 ) elements. The exponent 3 / 4 is optimal. This extends to the cubic lattice analogous results that have already been established for the triangular, the hexagonal, and the square lattice in two space dimensions.

中文翻译：

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$$N^{3/4}$$N3/4 三次晶格定律

我们通过强调其与 Wulff 形状在结晶问题中出现的关系来研究具有 n 个立方晶格元素的集合的边缘等周问题 (EIP)。边缘周长的最小化器 M n 显示为相对于它们的对称差异至多 O ( n 3 / 4 ) 个元素偏离相应的立方伍尔夫配置。指数 3 / 4 是最佳的。这扩展到立方晶格的类似结果，这些结果已经为三角形、六边形和方形晶格在二维空间中建立。