UTMS 2012 – 16 November 14 , 2012

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.