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N 3 / 4 Law in the Cubic Lattice
Edoardo Mainini1, Paolo Piovano2, Bernd Schmidt3
11Dipartimento di Ingegneria meccanica, energetica, gestionale e dei trasporti, Università degli studi di Genova, Via all'Opera Pia 15, 16145 Genoa, Italy.
None:
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 of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most 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.
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