Two-point dilation-homogeneous metric spaces

TL;DR

This paper classifies metric spaces with specific dilation properties, revealing that Euclidean spaces are uniquely characterized by these properties plus the existence of three collinear points.

Contribution

It provides a complete classification of two-point dilation-homogeneous metric spaces and characterizes Euclidean spaces through these properties and collinearity.

Findings

Classified all metric spaces with dilation properties.

Euclidean spaces uniquely characterized by these properties.

Identified conditions for metric collinearity in these spaces.

Abstract

The main aim of the paper is to give a full classification (up to isometry) of all metric spaces X with the following two properties: X contains a compact set with non-empty interior; and for any three distinct points a, b and c of X there exists a (bijective) dilation on X that fixes a and sends b to c. As a consequence, we obtain a new characterisation of the Euclidean spaces: these are (up to isometry) precisely all metric spaces that have the above two properties, and (in addition) contain three distinct points x, y, z that are metrically collinear (that is, for which d(x,z) = d(x,y)+d(y,z)).