A discrete isoperimetric inequality on lattices

Abstract

We establish an isoperimetric inequality with constraint by 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 Cabré (Butll Soc Catalana Mat 15:7-27, 2000) to prove the classical isoperimetric inequality.