  The aim of this paper is to investigate the differential geometry of immersed
surfaces in three-dimensional normed spaces from the viewpoint of affine
differential geometry. We endow the surface with a useful Riemannian metric
which is closely related to normal curvature, and from this we re-calculate the
Minkowski Gaussian and mean curvatures. These curvatures are also re-obtained
in terms of ambient affine distance functions, and as a consequence we
characterize minimal surfaces as the solutions of a certain differential
equation. We also investigate in which cases it is possible that the affine
normal and the Birkhoff normal vector fields of an immersion coincide, proving
that this only happens when the geometry is Euclidean.
