Equidistribution of Neumann data mass on simplices and a simple inverse problem

In this paper we study the behaviour of the Neumann data of Dirichlet eigenfunctions on simplices. We prove that the $L^2$ norm of the (semi-classical) Neumann data on each face is equal to $2/n$ times the $(n-1)$-dimensional volume of the face divided by the volume of the simplex. This is a generalization of \cite{Chr-tri} to higher dimensions. Again it is {\it not} an asymptotic, but an exact formula. The proof is by simple integrations by parts and linear algebra. We also consider the following inverse problem: do the {\it norms} of the Neumann data on a simplex determine a constant coefficient elliptic operator? The answer is yes in dimension 2 and no in higher dimensions.