  In this paper, we obtain optimal upper bounds for all the Neumann eigenvalues
in two situations (that are closely related). First we consider a
one-dimensional Sturm-Liouville eigenvalue problem where the density is a
function $h(x)$ whose some power is concave. We prove existence of a maximizer
for $\mu_k(h)$ and we completely characterize it. Then we consider the Neumann
eigenvalues (for the Laplacian) of a domain $\Omega\subset \mathbb{R}^d$ of
given diameter and we assume that its profile function (defined as the $d-1$
dimensional measure of the slices orthogonal to a diameter) has also some power
that is concave. This includes the case of convex domains in $\mathbb{R}^d$,
containing and generalizing previous results by P. Kr\"oger. On the other hand,
in the last section, we give examples of domains for which the upper bound
fails to be true, showing that, in general, $\sup D^2(\Omega)\mu_k(\Omega)=
+\infty$.
