A Discrete Sobolev Inequality on Lattices

Recently Hamamuki proved an optimal isoperimetric inequality with constraint in an \(n\)-dimensional lattice. The main idea of the proof is based on the ABP method, which was used by Cabré to give a new proof of the classical isoperimetric inequality. In this paper we generalize Hamamuki’s isoperimetric inequality to a Sobolev inequality by using Brendle’s new ideas.