Title: The affine approach to homogeneous geodesics in homogeneous Finsler spaces,
Abstract: In a recent paper, it was claimed that any homogeneous Finsler space of odd
dimension admits a homogeneous geodesic through any point. For the proof, the
algebraic method dealing with the reductive decomposition of the Lie algebra of
the isometry group was used. However, the proof contains a serious gap. In the
present paper, homogeneous geodesics in Finsler homogeneous spaces are studied
using the affine method, which was developed in earlier papers by the author.
The mentioned statement is proved correctly and it is further proved that any
homogeneous Berwald space or homogeneous reversible Finsler space admits a
homogeneous geodesic through any point.