  We prove that the nonlinear part $H^{*}$ of the KdV Hamiltonian $H^{kdv}$,
when expressed in action variables $I = (I_{n})_{n\ge 1}$, extends to a real
analytic function on the positive quadrant $\ell^2_+(\mathbb N)$ of
$\ell^{2}(\mathbb N)$ and is strictly concave near $0$. As a consequence, the
differential of $H^{*}$ defines a local diffeomorphism near $0$ of
$\ell_{\mathbb C}^{2}(\mathbb N)$.
