The Best Constant of Discrete Sobolev Inequality on Regular Polyhedron(Theory)

Abstract

We obtained the best constant of discrete Sobolev inequality on regular M-hedron for M=4, 6, 8, 12, 20. Let N be the number of vertices, whose indices are suitably chosen to express the symmetries of regular polyhedrons. We introduce a graph Laplacian A on regular polyhedrons. A is N×N real symmetric non-negative definite matrix and has an eigenvalue 0, whose corresponding eigenvector is 1=^t(1,1,…,1). Introducing Sobolev energy E(u) and E(a;u), we have two kinds of discrete Sobolev inequalities [numerical formula] For these inequalities, we obtained the best constants.