Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On quasisymmetric mappings in semimetric spaces

Evgeniy Petrov, Ruslan Salimov

TL;DR

The paper extends the theory of quasisymmetric mappings to general semimetric spaces by introducing and employing triangle functions and $\eta$-quasisymmetry. It develops structural preservation results showing when such mappings retain triangle inequalities, Ptolemy's inequality, and metric betweenness, and it provides a generalized Tukia– Väisälä inequality for diameter ratios under semimetric distortions. It also connects quasisymmetry to weak similarities, establishing conditions under which these notions coincide or diverge, and offers concrete implications for isometric embeddability and related configurations. Overall, the work broadens the applicability of quasisymmetric concepts beyond metric spaces, with potential impacts on geometric analysis in semimetric contexts and related algorithmic considerations.

Abstract

The class of quasisymmetric mappings on the real axis was first introduced by A. Beurling and L. V. Ahlfors in 1956. In 1980 P. Tukia and J. Väisälä considered these mappings between general metric spaces. In our paper we generalize the concept of quasisymmetric mappings to the case of general semimetric spaces and study some properties of these mappings. In particular, conditions under which quasisymmetric mappings preserve triangle functions, Ptolemy's inequality and the relation ``to lie between'' are found. Considering quasisymmetric mappings between semimetric spaces with different triangle functions we have found a new estimation for the ratio of diameters of two subsets, which are images of two bounded subsets. This result generalizes the well-known Tukia-Väisälä inequality. Moreover, we study connections between quasisymmetric mappings and weak similarities which are a special class of mappings between semimetric spaces.

On quasisymmetric mappings in semimetric spaces

TL;DR

The paper extends the theory of quasisymmetric mappings to general semimetric spaces by introducing and employing triangle functions and -quasisymmetry. It develops structural preservation results showing when such mappings retain triangle inequalities, Ptolemy's inequality, and metric betweenness, and it provides a generalized Tukia– Väisälä inequality for diameter ratios under semimetric distortions. It also connects quasisymmetry to weak similarities, establishing conditions under which these notions coincide or diverge, and offers concrete implications for isometric embeddability and related configurations. Overall, the work broadens the applicability of quasisymmetric concepts beyond metric spaces, with potential impacts on geometric analysis in semimetric contexts and related algorithmic considerations.

Abstract

The class of quasisymmetric mappings on the real axis was first introduced by A. Beurling and L. V. Ahlfors in 1956. In 1980 P. Tukia and J. Väisälä considered these mappings between general metric spaces. In our paper we generalize the concept of quasisymmetric mappings to the case of general semimetric spaces and study some properties of these mappings. In particular, conditions under which quasisymmetric mappings preserve triangle functions, Ptolemy's inequality and the relation ``to lie between'' are found. Considering quasisymmetric mappings between semimetric spaces with different triangle functions we have found a new estimation for the ratio of diameters of two subsets, which are images of two bounded subsets. This result generalizes the well-known Tukia-Väisälä inequality. Moreover, we study connections between quasisymmetric mappings and weak similarities which are a special class of mappings between semimetric spaces.
Paper Structure (8 sections, 33 theorems, 121 equations, 1 figure)

This paper contains 8 sections, 33 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some properties of semimetric spaces and quasisymmetric mappings
  3. Properties of quasisymmetric mappings to preserve the structures of spaces
  4. Property to preserve triangle inequalities
  5. Ptolemy's inequality preserving mappings
  6. Metric betweenness preserving mappings
  7. A generalization of the Tukia-Väisälä inequality
  8. Connections with weak similarities

Key Result

Proposition 2.3

Let $(X,d)$ be a semimetric space. If every closed ball of $(X,d)$ is a closed subset of the topological space $(X,\tau_d)$, then, for all points $x,y \in X$ and each sequence $(x_n)_{n\in \mathbb{N}}$, the double equality implies that $x=y$.

Figures (1)

  • Figure 1: The metric space $(X,d)$.

Theorems & Definitions (74)

  • Definition 1.1
  • Definition 1.2
  • Example 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 64 more