  Anisotropic generalized Orlicz spaces have been investigated in many recent
papers, but the basic assumptions are not as well understood as in the
isotropic case. We study the greatest convex minorant of anisotropic
$\Phi$-functions and prove the equivalence of two widely used conditions in the
theory of generalized Orlicz spaces, usually called (A1) and (M). This provides
a more natural and easily verifiable condition for use in the theory of
anisotropic generalized Orlicz spaces for results such as Jensen's inequality
which we obtain as a corollary.
