Determine the optimal aspect-ratio threshold

Determine the optimal threshold for the aspect ratio of a minimum-area enclosing rectangle that guarantees the lower bound bc42(a9) 2c0b2/Lb2 for a bounded open convex planar domain a9, where L is the longer side of the rectangle.

Background

Theorem 1.1 proves that if the shorter-to-longer side ratio W/L of a minimum-area enclosing rectangle is at most c1 9 0.443485, then the first nonzero Neumann eigenvalue satisfies bc42(a9) 2 c0b2/Lb2, with equality only for the rectangle itself. Appendix A constructs trapezoidal counterexamples for every ratio r in (ra0, 1/2), where ra0 9 0.457993, showing that the theorem cannot hold with any threshold larger than ra0. Consequently, the optimal threshold is known only to lie in the narrow interval [c1, ra0].