Two-point dilation-homogeneous metric spaces

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)).