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Sums of Laplace eigenvalues --- rotations and tight frames in higher dimensions

The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among simplexes for the regular simplex (the regular tetrahedron, in three dimensions), maximal among parallelepipeds for the hypercube, and maximal among ellipsoids for the ball, provided the volume and moment of inertia of an "inverse" body are suitably normalized. This result holds for Dirichlet, Robin and Neumann eigenvalues. Additionally, the cubical torus is shown to be maximal among flat tori. The method of proof involves tight frames for euclidean space generated by the orbits of the rotation group of the extremal domain. The ball is conjectured to maximize sums of Neumann eigenvalues among general convex domains, provided the volume and moment of inertia of the polar dual of the domain are suitably normalized.