  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
$\{d_{\mathrm{GH}}^{\scriptscriptstyle{(p)}}\}_{p\in[1,\infty]}$.
