Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24492220210901Rank inequality in homogeneous Finsler geometry171184445410.22060/ajmc.2021.20210.1058ENMingXuSchool of Mathematical Sciences, Capital Normal University, Beijing 100048, P.R. ChinaJournal Article20210627This is a survey on some recent progress in homogeneous Finsler geometry. Three topics are discussed, the classification of positively curved homogeneous Finsler spaces, the geometric and topological properties of homogeneous Finsler spaces satisfying $K\geq0$ and the (FP) condition, and the orbit number of prime closed geodesics in a compact homogeneous Finsler manifold. These topics share the same similarity that the same rank inequality, i.e., $\mathrm{rank}G\leq\mathrm{rank}H+1$ for $G/H$ with compact $G$ and $H$, plays an important role. In this survey, we discuss in each topic how the rank inequality is proved, explain its importance, and summarize some relevant results.https://ajmc.aut.ac.ir/article_4454_7c0a80121aaf6969c50902c807a75add.pdf