  In this paper we prove that among all convex domains of the plane with two
axis of symmetry, the maximizer of the first non trivial Neumann eigenvalue
$\mu_1$ with perimeter constraint is achieved by the square and the equilateral
triangle. Part of the result follows from a new general bound on $\mu_1$
involving the minimal width over the area. Our main result partially answers to
a question addressed in 2009 by R. S. Laugesen, I. Polterovich, and B. A.
Siudeja.
