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}+mathrm{o}(n^{3/4}) points. The constant K_t is explicitly determined and shown to be sharp.
Highlights
This paper is concerned with the edge-isoperimetric problem (EIP) in the triangular lattice
The edge perimeter | (Cn)| of a set Cn ∈ Cn is the cardinality of the edge boundary of
We provide a first characterization of the minimizers Mn of the EIP by introducing an isoperimetric inequality in terms of suitable notions of area and perimeter of configurations in Cn and by showing that the connected minimizers Mn of the EIP are optimal with respect to it
Summary
This paper is concerned with the edge-isoperimetric problem (EIP) in the triangular lattice. We observe that in view of Theorem 1.1 estimates (15) – (18) and (20) provide a measure in different topologies of the fluctuation of the isoperimetric configurations in Lt with respect to corresponding maximal hexagons. In lower bound for fact, a sequence the radius rMn of minimizers established in Mn satisfying the proof (24) with equality can be explicitly constructed Note that such configurations Mn are singled out among configurations that present extra elements outside their maximal hexagon. We observe that analogous results to Theorem 1.2 were obtained in the context of the crystallization problem in the square lattice in Mainini et al (2014a, b) with a substantially different method (even though based on an isoperimetric characterization of the minimizers) resulting only in suboptimal estimates.
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