Global and Local Bounds on the Fundamental Ratio of Triangles and Quadrilaterals

We present a new, computer-assisted, proof that for all triangles in the plane, the equilateral triangle uniquely maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues. This proves an independent proof the triangular Ashbaugh-Benguria-Payne-Polya-Weinberger conjecture first proved in arXiv:0707.3631 [math.SP] and arXiv:2009.00927 [math.SP]. Inspired by arXiv:1109.4117 [math.SP], the primary method is to use a perturbative estimate to determine a local optimum, and to then use a continuity estimate for the fundamental ratio to perform a rigorous computational search of parameter space. We repeat a portion of this proof to show that the square is a strict local optimizer of the fundamental ratio among quadrilaterals in the plane