  For a given bounded Lipschitz set $\Omega$, we consider a Steklov--type
eigenvalue problem for the Laplacian operator whose solutions provide extremal
functions for the compact embedding $H^1(\Omega)\hookrightarrow L^2(\partial
\Omega)$. We prove that a conjectured reverse Faber--Krahn inequality holds
true at least in the class of Lipschitz sets which are "close" to a ball in a
Hausdorff metric sense. The result implies that among sets of prescribed
measure, balls are local minimizers of the embedding constant.
