On the Convexity of the KdV Hamiltonian

We prove that the nonlinear part H∗ of the KdV Hamiltonian Hkdv, when expressed in action variables I=(In)n≥1, extends to a real analytic function on the positive quadrant l2+(N) of l2(N) and is strictly concave near 0. As a consequence, the differential of H∗ defines a local diffeomorphism near 0 of l2C(N).