Quantitative Kröger inequalities for Neumann eigenvalues of convex domains

Refining the sharp upper bounds ${\mu }_{k,d}^{\ast }$ obtained by Kröger (1999) for the $k$-th Neumann eigenvalue of a convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a constant $C(k,d)>0$ such that $$D_{\mathrm{\Omega }}^2{\mu }_k(\mathrm{\Omega })\le {\mu }_{k,d}^{\ast }-C(k,d)a_2(\mathrm{\Omega }{)}^2\mathrm{/}{D}_{\mathrm{\Omega }}^2$$ where $D_{\mathrm{\Omega }}$ is the diameter of $\mathrm{\Omega }$ and $a_2(\mathrm{\Omega })$ is the second largest semiaxis of the John ellipsoid of $\mathrm{\Omega }$. In the planar case, for $k=1$ we also give an explicit value of the constant $C(1,2)$.