Optimal dependence on the generalized inradius fraction in Lieb/Davies-type bounds

Determine whether the Ψ_r^{-2/d} blow-up as Ψ_r→0 and the linear dependence on (1−Ψ_r) as Ψ_r→1 are optimal in lower bounds of the form λ^{D}_Ω ≥ c r^{-2}·f(Ψ_r) for the first Dirichlet eigenvalue of the Laplacian on open subsets of R^d, where Ψ_r:=sup_{x∈Ω}|Ω∩B_r(x)|/|B_r(x)|.

Background

The paper derives bounds exhibiting specific quantitative behaviors: divergence of the lower bound as Ψ_r→0 and linear vanishing as Ψ_r→1. Whether these rates are sharp across all domains is unknown.

Clarifying optimality would calibrate the precise role of the generalized inradius fraction in spectral lower bounds and potentially guide improvements or establish limits of current techniques.