A note on generalized four-point inequality

TL;DR

This paper explores a generalized four-point inequality in semimetric spaces, extending known metric space inequalities and analyzing the preservation of this inequality under specific mappings.

Contribution

It introduces a new generalized four-point inequality in semimetric spaces and studies its invariance under quasisymmetric and quasimöbius mappings.

Findings

Established conditions for preservation of the inequality under mappings

Extended known metric inequalities to semimetric spaces

Provided theoretical framework for generalized inequalities

Abstract

In 2017 M. Bessenyei and Z. P\'ales introduced a definition of a triangle function which 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 the six distances determined by four points and introduce a definition of a generalized four-point inequality in semimetric spaces. Conditions under which quasisymmetric mappings and quasim\"{o}bius mappings between semimetric spaces preserve such inequality are obtained.