  We analyze the influence of the magnetic field in the convexity properties of
the relativistic magnetohydrodynamics system of equations. To this purpose we
use the approach of Lax, based on the analysis of the linearly
degenerate/genuinely non-linear nature of the characteristic fields. Degenerate
and non-degenerate states are discussed separately and the non-relativistic,
unmagnetized limits are properly recovered. The characteristic fields
corresponding to the material and Alfv\'en waves are linearly degenerate and,
then, not affected by the convexity issue. The analysis of the characteristic
fields associated with the magnetosonic waves reveals, however, a dependence of
the convexity condition on the magnetic field. The result is expressed in the
form of a generalized fundamental derivative written as the sum of two terms.
The first one is the generalized fundamental derivative in the case of purely
hydrodynamical (relativistic) flow. The second one contains the effects of the
magnetic field. The analysis of this term shows that it is always positive
leading to the remarkable result that the presence of a magnetic field in the
fluid reduces the domain of thermodynamical states for which the EOS is
non-convex.
