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 Poincaré inequality for triangles is derived.
The proof relies on symmetry of the Neumann fundamental mode for isosceles
triangles with aperture less than π/3. Antisymmetry is proved for apertures
greater than π/3.
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