Sharpness of the constant in Kovařík’s Robin inradius bound

Determine whether the constant 1/4 in Kovařík’s lower bound λ^{R,σ}_Ω ≥ (1/4)·σ/(R_Ω(1+σ R_Ω)) for the first eigenvalue of the Robin Laplacian with positive constant σ on convex domains is optimal.

Background

Kovařík established an inradius-based lower bound for the first Robin eigenvalue with a positive parameter σ, analogous to inradius bounds for Dirichlet problems. However, its optimality is unsettled.

The authors note they extend such bounds to mean-convex sets and improve constants in certain regimes, yet the sharpness of the original 1/4 coefficient remains an unresolved question.