The convex-domain bounds conjecture for the Dirichlet fundamental tone

Let $\Omega$ be a bounded convex plane domain, with area $A$, moment of inertia $I$, and first Dirichlet Laplace eigenvalue $\lambda_1$. Convex-domain bounds conjecture.

$$\frac{9}{2}\pi^2<\left.\lambda_1\frac{A^3}{I}\right|_{\Omega}\leq12\pi^2.$$

Equality in the upper bound should occur for equilateral triangles and all rectangles, while the lower bound should be approached asymptotically by degenerate acute isosceles triangles and sectors. This proposes sharp universal bounds and identifies both equality and limiting configurations; the paper does not report a proof in general.