  We investigate the relations between normalized critical points of the
nonlinear Schr\"odinger energy functional and critical points of the
corresponding action functional on the associated Nehari manifold. Our first
general result is that the ground state levels are strongly related by the
following duality result: the (negative) energy ground state level is the
Legendre-Fenchel transform of the action ground state level. Furthermore,
whenever an energy ground state exists at a certain frequency, then all action
ground states with that frequency have the same mass and are energy ground
states too. We prove that the converse is in general false and that the action
ground state level may fail to be convex. Next we analyze the differentiability
of the ground state action level and we provide an explicit expression
involving the mass of action ground states. Finally we show that similar
results hold also for local minimizers.
