For any convex domain Ω in R^d and any k, D_Ω² μ_k(Ω) ≤ μ_{k,d}^* minus a positive term C(k,d) times a_2(Ω)² over D_Ω², where a_2 is the second semiaxis of the John ellipsoid.
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Quantitative Kr\"{o}ger inequalities for Neumann eigenvalues of convex domains
For any convex domain Ω in R^d and any k, D_Ω² μ_k(Ω) ≤ μ_{k,d}^* minus a positive term C(k,d) times a_2(Ω)² over D_Ω², where a_2 is the second semiaxis of the John ellipsoid.