Sharp ratios for low-index Neumann eigenvalues on convex domains

Let $\Omega\subset\mathbb{R}^N$ be a bounded open convex set, and let $0=\mu_0(\Omega)<\mu_1(\Omega)\le \mu_2(\Omega)\le\cdots$ be the Neumann eigenvalues of the Laplacian, repeated according to multiplicity. We prove the sharp bounds $$\mu_2(\Omega)\le 4\mu_1(\Omega),\qquad \mu_3(\Omega)\le 9\mu_1(\Omega).$$ The first estimate resolves a problem attributed to Henrot, while the second gives the next sharp case predicted by the one-dimensional model. The constants are optimal in every dimension.