On similarly homogeneous but not homogeneous spaces with inner metric and their similarity groups (Q2799785)

A bijection of a metric space \(X\) onto itself is called a similarity if it changes the distance of points by a constant factor. \(X\) is called similarly homogeneous if the action of the group \(G\) of all similarities on \(X\) is transitive. Without proof, for a similarly homogeneous but not homogeneous space conditions are given implying that \(G\) is the semidirect product of \(\mathbb{R}_+\) with the isometric group of \(X\) and that \(X\) is homeomorphic to \(\mathbb{R}_+\times c^{-1}(a)\), where \(a\in \mathbb R_+\) and \(c:X\to\mathbb{R}_+\) is the so-called radius of completeness, defined as the upper bound of all numbers \(r\) such that the ball \(B(x,r)\) is complete.NEWLINENEWLINEFor the entire collection see [Zbl 1298.53003].