Quantitative Kr\"{o}ger inequalities for Neumann eigenvalues of convex domains

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