  In this paper, we address the problem of maximizing the Steklov eigenvalues
with a diameter constraint. We provide an estimate of the Steklov eigenvalues
for a convex domain in terms of its diameter and volume and we show the
existence of an optimal convex domain. We establish that balls are never
maximizers, even for the first non-trivial eigenvalue that contrasts with the
case of volume or perimeter constraints. Under an additional regularity
assumption, we are able to prove that the Steklov eigenvalue is multiple for
the optimal domain. We illustrate our theoretical results by giving some
optimal domains in the plane thanks to a numerical algorithm.
