A sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues

We prove a sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues of bounded Lipschitz domains in ${\mathbb{R}}^N$. More precisely, if $N\ge 2$ and $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded Lipschitz domain, then $$\prod _{j=1}^N{\sigma }_j(\mathrm{\Omega })\le \frac{{\omega }_N}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }},$$ where $0={\sigma }_0(\mathrm{\Omega })<{\sigma }_1(\mathrm{\Omega })\le {\sigma }_2(\mathrm{\Omega })\le \cdots$ are the Steklov eigenvalues of $\mathrm{\Omega }$, and ${\omega }_N$ denotes the volume of the unit ball in ${\mathbb{R}}^N$. This extends the convex-domain theorem of Henrot, Philippin, and Safoui to arbitrary bounded Lipschitz domains, and in particular settles the remaining higher-dimensional case of a problem posed by Henrot.