  We consider the Dirichlet eigenvalue problem on a simple polytope. We use the
Rellich identity to obtain an explicit formula expressing the Dirichlet
eigenvalue in terms of the Neumann data on the faces of the polytope of the
corresponding eigenfunction. The formula is particular simple for polytopes
admitting an inscribed ball tangent to all the faces. Our result could be
viewed as a generalization of similar identities for simplices recently found
by Christianson [1][2].
