A Discrete Isoperimetric Inequality on Lattices

Abstract

Abstract. We establish an isoperimetric inequality with constraint by ndimensional lattices. We prove that, among all domains which consist of rectangular parallelepipeds with the common side-lengths, a cube is the best shape to minimize the ratio involving its perimeter and volume as long as the cube is realizable by the lattice. For its proof a solvability of finite difference PoissonNeumann problems is verified. Our approach to the isoperimetric inequality is based on the technique used in a proof of the Aleksandrov-Bakelman-Pucci maximum principle, which was originally proposed by Cabré in 2000 to prove the classical isoperimetric inequality.

Sharp discrete isoperimetric inequalities in periodic graphs via discrete PDE and Semidiscrete Optimal Transport

“…In the discrete case the Cabré setup was generalized by [32] in the case of orthogonal product lattices, however we find that for the general case a semidiscrete optimal transport interpretation is more instructive.…”

“…However, virtually all versions of the inequalities do not treat the question of determining the isoperimetric shapes, or discussing cases in which equality in the isoperimetric inequality can be reached. The only exception which we are aware of is the paper [32], which treats the case of a lattice V = DZ d , with D = diag(λ 1 , . .…”

“…The setup in that case is based on the Cabré method from (1.7). The maximum principle tools from [32] can be seen as fitting within the broader theory of discretization of PDE tools, for which we refer to the above work for a broad list of references, and here we only mention the works of Merkov [45] and the in-depth treatment by Kuo and Trudinger, see e.g. [38], [39], [40] and the references therein.…”

“…In [32] the question of correctly geneneralizing the result to obtain isoperimetric constants in other lattices is mentioned as an interesting open problem at the end of their paper. This is the main motivation and direction of the present work.…”

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We develop criteria based on a calibration argument via discrete PDE and semidiscrete optimal transport, for finding sharp isoperimetric inequalities of the form (♯Ωwhere Ω is a subset of vertices of a graph and − → ∂Ω is the oriented edge-boundary of Ω, as well as the optimum isoperimetric shapes Ω. The method is a discrete counterpart to Optimal Transport and ABP method proofs valid in the continuum, and answers a question appearing in Hamamuki [32], extending that work valid for rectangular grids, to a larger class of graphs, including graphs dual to simplicial meshes of equal volume. We also connect the problem to the theory Voronoi tessellations and of Aleksandrov solutions from semidiscrete optimal transport.

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