Upper bound for the first $p$-Steklov eigenvalue in ${\mathbb{R}}^n$

For $p\in (1,n]$ and any bounded convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, we prove the sharp inequality $${\mathrm{\Lambda }}_p(\mathrm{\Omega }):=\frac{W_p(\mathrm{\Omega })}{P(\mathrm{\Omega })V(\mathrm{\Omega }{)}^{p\mathrm{/}n}}\ge {\omega }_n^{-p\mathrm{/}n},W_p(\mathrm{\Omega })={\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }x{\mathrm{\mid }}^{p}dS,$$ with equality holding exactly at centered balls. Combining this with the isoperimetric inequality yields the explicit upper bound $${\sigma }_{1,p}(\mathrm{\Omega })\le \frac{A(n,p)}{r({\mathrm{\Omega }}^{\ast }{)}^{p-1}},$$ where ${\mathrm{\Omega }}^{\ast }$ is a ball having the same perimeter as $\mathrm{\Omega }$, and $A(n,p)=1$ for $1<p\le 2$, $A(n,p)=n^{p\mathrm{/}2-1}$ for $2<p\le n$. When $p=2$, the result recovers the higher-dimensional Weinstock inequality of Bucur et al. [J. Differential Geom. 2021]. We also obtain an explicit upper bound for the first Wentzell eigenvalue of the $p$-Laplacian on convex domains.