A Remark on the Steklov–Poincaré Inequality

In an $n$-dimensional bounded domain $\Omega_n$, $n\ge 2$, we prove the Steklov–Poincaré inequality with the best constant in the case where $\Omega_n$ is an $n$-dimensional ball. We also consider the case of an unbounded domain with finite measure, in which the Steklov–Poincaré inequality is proved on the basis of a Sobolev inequality.