A Discrete Isoperimetric Inequality on Lattices

Abstract

We establish an isoperimetric inequality with constraint by $$n$$n-dimensional lattices. We prove that, among all sets which consist of lattice translations of a given rectangular parallelepiped, 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 Poisson---Neumann 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 Cabre (Butll Soc Catalana Mat 15:7---27, 2000) to prove the classical isoperimetric inequality.