[Formula: see text] Law in the Cubic Lattice

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 [Formula: see text] of the edge perimeter are shown to deviate from a corresponding cubic Wul...

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Detalles Bibliográficos
Autores principales: Mainini, Edoardo, Piovano, Paolo, Schmidt, Bernd, Stefanelli, Ulisse
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2019
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6733839/
https://www.ncbi.nlm.nih.gov/pubmed/31555015
http://dx.doi.org/10.1007/s10955-019-02350-z
Descripción
Sumario: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 [Formula: see text] of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most [Formula: see text] 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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