  The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be
maximal among triangles for the equilateral triangle, maximal among
parallelograms for the square, and maximal among ellipses for the disk,
provided the ratio $\text{(area)}^3/\text{(moment of inertia)}$ for the domain
is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and
similar conclusions are derived for Robin boundary conditions and Schr\"odinger
eigenvalues of potentials that grow at infinity. A key ingredient in the method
is the tight frame property of the roots of unity. For general convex plane
domains, the disk is conjectured to maximize sums of Neumann eigenvalues.
