A note on generalized four-point inequality

Abstract

In 2017, M. Bessenyei and Z. Páles [1] introduced a definition of a triangle function that generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept, we discuss already known inequalities in metric spaces that relate six distances determined by four points and introduce a definition of the generalized four-point inequality in semimetric spaces. Conditions under which quasisymmetric mappings and quasi-Möbius mappings between semimetric spaces preserve such an inequality are obtained.