  We prove three facts about intrinsic geometry of surfaces in a normed
(Minkowski) space. When put together, these facts demonstrate a rather
intriguing picture. We show that (1) geodesics on saddle surfaces (in a space
of any dimension) behave as they are expected to: they have no conjugate points
and thus minimize length in their homotopy class; (2) in contrast, every
two-dimensional Finsler manifold can be locally embedded as a saddle surface in
a 4-dimensional space; and (3) geodesics on convex surfaces in a 3-dimensional
space also behave as they are expected to: on a complete strictly convex
surface, no complete geodesic minimizes the length globally.
