What is the real meaning of the Froissart theorem ?

Abstract: The Froissart bounds for amplitudes and cross sections are explained and reconsidered to clarify the role of different assumptions. It is the physical conditions of unitarity and of no massless exchanges, together with mathematical properties of the Legendre functions, that imply much softer high-energy asymptotics for elastic amplitudes at physical angles than for the same amplitudes at nonphysical angles. The canonical log-squared boundary for \sigma_{tot} appears only under the additional hypothesis that the amplitude at any nonphysical angle cannot grow faster than some power of energy. The Froissart results are further shown to admit some reinforcement. Comparison of the familiar and new Froissart-like restrictions with the existing data on \sigma_{tot} and diffraction slope at all available energies (including LHC) does not allow yet to unambiguously determine the asymptotic behavior of \sigma_{tot}, but shows that its current increase cannot be saturated (i.e., maximally rapid).