Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality

The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincar'{e} inequality for triangles is derived. The proof relies on symmetry of the Neumann fundamental mode for isosceles triangles with aperture less than $\pi \mathrm{/}3$. Antisymmetry is proved for apertures greater than $\pi \mathrm{/}3$.