  We construct a class of Finsler metrics in three-dimensional space such that
all their geodesics are lines, but not all planes are extremal for their
Hausdorff area functionals. This shows that if the Hausdorff measure is used as
notion of volume on Finsler spaces, then totally geodesic submanifolds are not
necessarily minimal, filling results such as those of S. Ivanov do not hold,
and integral-geometric formulas do not exist. On the other hand, using the
Holmes-Thompson definition of volume, we prove a general Crofton formula for
Finsler spaces and give an easy proof that their totally geodesic hypersurfaces
are minimal.
