*Accepted Paper*

**Inserted:** 14 dec 2015

**Last Updated:** 7 nov 2016

**Journal:** Journal of Nonlinear Science

**Year:** 2016

**Abstract:**

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 $K_t\, n^{3/4}+{\rm o}(n^{3/4})$ points. The constant $K_t$ is explicitly determined and shown to be sharp.

**Keywords:**
isoperimetric inequality, wulff shape, triangular lattice, Edge-Isoperimetric Problem, Edge-perimeter, $N^{3/4}$ law

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