Sharp ratios for low-index Neumann eigenvalues on convex domains

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ be a bounded open convex set, and let $0={\mu }_0(\mathrm{\Omega })<{\mu }_1(\mathrm{\Omega })\le {\mu }_2(\mathrm{\Omega })\le \cdots$ be the Neumann eigenvalues of the Laplacian, repeated according to multiplicity. We prove the sharp bounds $${\mu }_2(\mathrm{\Omega })\le 4{\mu }_1(\mathrm{\Omega }),{\mu }_3(\mathrm{\Omega })\le 9{\mu }_1(\mathrm{\Omega })\mathrm{.}$$ 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.