  We prove that among all triangles of given diameter, the equilateral triangle
minimizes the sum of the first $n$ eigenvalues of the Neumann Laplacian, when
$n \geq 3$.
  The result fails for $n=2$, because the second eigenvalue is known to be
minimal for the degenerate acute isosceles triangle (rather than for the
equilateral) while the first eigenvalue is 0 for every triangle. We show the
third eigenvalue is minimal for the equilateral triangle.
