Abstract
In this paper we prove that among all convex domains of the plane with two axes of symmetry, the maximizer of the first non-trivial Neumann eigenvalue μ 1 \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 μ 1 \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.
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