Minimum-Area Enclosing Rectangles and the First Neumann Eigenvalue of Planar Convex Domains

Let $\mathrm{\Omega }$ be a bounded convex domain in the plane and let $R$ be a minimum-area rectangle containing $\bar{\mathrm{\Omega }}$. We prove that if the ratio of the shorter side of $R$ to its longer side is at most $\rho \approx 0.443485$, then the first nonzero Neumann eigenvalue of $\mathrm{\Omega }$ is bounded from below by that of $R$. Moreover, equality holds if and only if $\mathrm{\Omega }$ coincides with the interior of $R$. The proof is based on a flux identity for the means of a test function over vertical sections of $\mathrm{\Omega }$, combined with weighted estimates involving the distributional second derivatives of the boundary functions. Finally, we show that the threshold $\rho \approx 0.443485$ is nearly optimal by constructing, for every ratio in $(0.457994,1\mathrm{/}2)$, trapezoids that are arbitrarily close to their minimum-area enclosing rectangle and for which the inequality fails.