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We consider the Laplace operator on a triangle, subject to attractive Robinboundary conditions. We prove that the equilateral triangle is a localmaximiser of the lowest eigenvalue among all triangles of a given area providedthat the negative boundary parameter is sufficiently small in absolute value,with the smallness depending on the area only. Moreover, using various trialfunctions, we obtain sufficient conditions for the global optimality of theequilateral triangle under fixed area constraint in the regimes of small andlarge couplings. We also discuss the constraint of fixed perimeter.

Reverse isoperimetric inequality for the lowest Robin eigenvalue of a triangle