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We consider the Dirichlet eigenvalue problem on a simple polytope. We use theRellich identity to obtain an explicit formula expressing the Dirichleteigenvalue in terms of the Neumann data on the faces of the polytope of thecorresponding eigenfunction. The formula is particular simple for polytopesadmitting an inscribed ball tangent to all the faces. Our result could beviewed as a generalization of similar identities for simplices recently foundby Christianson [1][2].

Rellich-Christianson type identities for the Neumann data mass of Dirichlet eigenfunctions on polytopes