  In this paper we prove the existence of a maximum for the first
Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical
hole under volume constraint. More precisely, if $\Omega=\Omega_0 \setminus
\bar{B}_{R_1}$, 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 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.
