  We consider the problem of minimising the $k$-th eigenvalue of the Laplacian
with some prescribed boundary condition over collections of convex domains of
prescribed perimeter or diameter. It is known that these minimisation problems
are well-posed for Dirichlet eigenvalues in any dimension $d\geq 2$ and any
sequence of minimisers converges to the ball of unit perimeter or diameter
respectively as $k\to +\infty$. In this paper, we show that the same is true in
the case of Neumann eigenvalues under diameter constraint in any dimension and
under perimeter constraint in dimension $d=2$. We also consider these problems
for mixed Dirichlet-Neumann eigenvalues, under an additional geometric
constraint, and discuss some applications of our proof techniques.
