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In this paper, we obtain optimal upper bounds for all the Neumann eigenvaluesin two situations (that are closely related). First we consider aone-dimensional Sturm-Liouville eigenvalue problem where the density is afunction $h(x)$ whose some power is concave. We prove existence of a maximizerfor $\mu_k(h)$ and we completely characterize it. Then we consider the Neumanneigenvalues (for the Laplacian) of a domain $\Omega\subset \mathbb{R}^d$ ofgiven 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 powerthat 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 boundfails to be true, showing that, in general, $\sup D^2(\Omega)\mu_k(\Omega)=+\infty$.

Optimal bounds for Neumann eigenvalues in terms of the diameter