On metric structures of normed gyrogroups

In this article, we indicate that the open unit ball in $n$-dimensional Euclidean space ${\mathbb{R}}^n$ admits norm-like functions compatible with the Poincar'e and Beltrami$-$Klein metrics. This leads to the notion of a normed gyrogroup, similar to that of a normed group in the literature. We then examine topological and geometric structures of normed gyrogroups. In particular, we prove that the normed gyrogroups are homogeneous and form left invariant metric spaces and derive a version of the Mazur$-$Ulam theorem. We also give certain sufficient conditions, involving the right-gyrotranslation inequality and Klee’s condition, for a normed gyrogroup to be a topological gyrogroup.