A characteristic of gyroisometries in Möbius gyrovector spaces

Abstract

Hugo Steinhaus (1966a, b) has asked whether inside each acute angled triangle there is a point from which perpendiculars to the sides divide the triangle into three parts of equal areas. In this paper, we prove that \(f:\mathbb{D} \rightarrow \mathbb{D}\) is a gyroisometry (hyperbolic isometry) if, and only if it is a continuous mapping that preserves the partition of a gyrotriangle (hyperbolic triangle) asked by Hugo Steinhaus.