  We prove that the non-linear part of the Hamiltonian of the KdV equation on
the circle, written as a function of the actions, defines a continuous convex
function on the $\ell^2$ space and derive for it lower and upper bounds in
terms of some functions of the $\ell^2$-norm. The proof is based on a new
representation of the Hamiltonian in terms of the quasimomentum and its
analysis using the conformal mapping theory.
