  The normal map given by Birkhoff orthogonality yields extensions of
principal, Gaussian and mean curvatures to surfaces immersed in
three-dimensional spaces whose geometry is given by an arbitrary norm and which
are also called Minkowski spaces. We obtain characterizations of the Minkowski
Gaussian curvature in terms of surface areas, and respective generalizations of
the classical theorems of Huber, Willmore, Alexandrov, and
Bertrand-Diguet-Puiseux are derived. A generalization of Weyl's formula for the
volume of tubes and some estimates for volumes and areas in terms of curvature
are obtained, and in addition we discuss also two-dimensional subcases of the
results in more detail.
