On $p$-metric spaces and the $p$-Gromov-Hausdorff distance

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 $\{d_{\mathrm{GH}}^{(p)}{\}}_{p\in [1,\infty ]}$ on ${\mathcal{M}}_p$ and study geometric and topological properties of these distances as well as the stability of certain canonical projections ${\mathfrak{S}}_p:\mathcal{M}\to {\mathcal{M}}_p$. 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 $\{d_{\mathrm{GH}}^{(p)}{\}}_{p\in [1,\infty ]}$.

The previous version of this paper is split into two papers: (1) the computational part of the previous version was expanded and written as a new paper which can be found at arXiv:2110.03136; (2) the current version of this paper contains the theoretical part of the previous version