  In this paper we prove the existence of a maximum for the first
Steklov-Dirichlet Laplacian eigenvalue in the class of convex sets with a fixed
spherical hole under volume constraint. More precisely, if $\Omega=\Omega_0
\setminus \overline{B}_{R_1}$, where $B_{R_1}$ is the ball centered at the
origin with radius $R_1>0$ and $\Omega_0\subset\mathbb{R}^n$, $n\geq 2$, is an
open bounded and convex set such that $B_{R_1}\Subset \Omega_0$, then the first
Steklov-Dirichlet Laplacian eigenvalue $\sigma_1(\Omega)$ has a maximum when
$R_1$ and the measure of $\Omega$ are fixed. Moreover, if $\Omega_0$ is
contained in a suitable ball, we prove that the spherical shell is the maximum.
