Abstract

For each given \(p\in[1,\infty]\) we investigate certain sub-family \(\mathcal{M}_p\) of the collection of all compact metric spaces \(\mathcal{M}\) which are characterized by the satisfaction of a strengthened form of the triangle inequality which encompasses, for example, the strong triangle inequality satisfied by ultrametric spaces. We identify a one parameter family of Gromov-Hausdorff like distances \(\{d_{\mathrm{GH}}^{\scriptscriptstyle{(p)}}\}_{p\in[1,\infty]}\) on \(\mathcal{M}_p\) and study geometric and topological properties of these distances as well as the stability of certain canonical projections \(\mathfrak{S}_p:\mathcal{M}\rightarrow \mathcal{M}_p\). For the collection \(\mathcal{U}\) of all compact ultrametric spaces, which corresponds to the case \(p=\infty\) of the family \(\mathcal{M}_p\), we explore a one parameter family of interleaving-type distances and reveal their relationship with \(\{d_{\mathrm{GH}}^{\scriptscriptstyle{(p)}}\}_{p\in[1,\infty]}\).