  We investigate the interaction energy and the possibility of itinerant
ferromagnetism in a strongly interacting Fermi gas at zero temperature in the
absence of molecule formation. The interaction energy is obtained by summing
the perturbative contributions of Galitskii-Feynman type to all orders in the
gas parameter. It can be expressed by a simple phase space integral of an
in-medium scattering phase shift. In both three and two dimensions (3D and 2D),
the interaction energy shows a maximum before reaching the resonance from the
Bose-Einstein condensate side, which provides a possible explanation of the
experimental measurements of the interaction energy. This phenomenon can be
theoretically explained by the qualitative change of the nature of the binary
interaction in the medium. The appearance of an energy maximum has significant
effects on the itinerant ferromagnetism. In 3D, the ferromagnetic transition is
reentrant and itinerant ferromagnetism exists in a narrow window around the
energy maximum. In 2D, the present theoretical approach suggests that itinerant
ferromagnetism does not exist, which reflects the fact that the energy maximum
becomes much lower than the energy of the fully polarized state.
