Two-point dilation-homogeneous metric spaces
Piotr Niemiec
TL;DR
The paper classifies all locally compact metric spaces that are two-point dilation-homogeneous and contain a compact set with non-empty interior, showing that up to isometry these spaces must be one of $D_{\alpha,r}$, $F_{n,a,b}$, or $R_{n,\alpha}$. It proves type-1 spaces are ultrametric and completely classified by $(N,a,b)$ with canonical models $F_{n,a,b}$, and type-2 spaces are characterized via Tits’ two-transitive action to yield Euclidean-like geometries $(\mathbb{R}^n,d_e^{\alpha})$ for $\alpha\in(0,1]$, $n>0$. A key consequence is a new Euclidean-characterization: Euclidean spaces are exactly the two-point dilation-homogeneous spaces with a compact interior and a metrically collinear triple, together with the dilation-extendability and abelian-group structure for Heine-Borel spaces. The results connect dilation groups with ultrametric and local-field inspired models, providing explicit canonical constructions and guiding Principles for the geometry of dilations.
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)).