  Results are obtained for two minimization problems: $$I_k(c)=\inf
\{\lambda_k(\Omega): \Omega\ \textup{open, convex in}\ \mathbb{R}^m,\
\mathcal{T}(\Omega)= c \},$$ and $$J_k(c)=\inf\{\lambda_k(\Omega): \Omega\
\textup{quasi-open in}\ \mathbb{R}^m, |\Omega|\le 1, \mathcal {P}(\Omega)\le c
\},$$ where $c>0$, $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the
Dirichlet Laplacian acting in $L^2(\Omega)$, $|\Omega|$ denotes the Lebesgue
measure of $\Omega$, $\mathcal{P}(\Omega)$ denotes the perimeter of $\Omega$,
and where $\mathcal{T}$ is in a suitable class set functions. The latter
include for example the perimeter of $\Omega$, and the moment of inertia of
$\Omega$ with respect to its center of mass.
