  In this paper we look for the domains minimizing the h-th eigenvalue of the
Dirichlet-Laplacian $\lambda$ h with a constraint on the diameter. Existence of
an optimal domain is easily obtained, and is attained at a constant width body.
In the case of a simple eigenvalue, we provide non standard (i.e., non local)
optimality conditions. Then we address the question whether or not the disk is
an optimal domain in the plane, and we give the precise list of the 17
eigenvalues for which the disk is a local minimum. We conclude by some
numerical simulations showing the 20 first optimal domains in the plane.
