We investigate the edge-isoperimetric problem (EIP) for sets of n points in the triangular lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. By introducing a suitable notion of perimeter and area, EIP minimizers are characterized as extremizers of an isoperimetric inequality: they attain maximal area and minimal perimeter among connected configurations. The maximal area and minimal perimeter are explicitly quantified in terms of n. In view of this isoperimetric characterizations, EIP minimizers \(M_n\) are seen to be given by hexagonal configurations with some extra points at their boundary. By a careful computation of the cardinality of these extra points, minimizers \(M_n\) are estimated to deviate from such hexagonal configurations by at most \(K_t\, n^{3/4}+\mathrm{o}(n^{3/4})\) points. The constant \(K_t\) is explicitly determined and shown to be sharp.

中文翻译：

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尖锐的[公式：参见文本]三角格上边等距问题最小化的定律。

我们通过强调晶格中与Wulff形状的出现之间的关系，来研究三角晶格中n个点的集合的边等距问题（EIP）。通过引入适当的周长和面积概念，EIP最小化器的特征是等距不等式的极值化器：它们在连接的配置中获得最大的面积和最小的周长。最大面积和最小周长根据n明确量化。考虑到这种等距特征，可以看到EIP最小化器\（M_n \）由六角形配置给出，在它们的边界处有一些额外的点。通过仔细计算这些额外点的基数，最小化器\（M_n \）估计最多偏离此类六边形配置\（K_t \，n ^ {3/4} + \ mathrm {o}（n ^ {3/4}）\）点。明确确定常数\（K_t \）并显示为尖锐的。