Geometry of Geometric Data Set II: Pyramid

The observable distance $d_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}$ based on measure concentration and the box distance $\mathrm{\square }$ based on collapsing theory are extended to geometric data sets introduced by Hanika–Schneider–Stumme. On the set $\mathcal{D}$ of isomorphism classes of geometric data sets, $d_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}$ is non-separable and $\mathrm{\square }$ is complete and non-separable. We introduce the class $\mathcal{D}\mathrm{/}\mathcal{L}$ of $\mathcal{L}$-compact geometric data sets in $\mathcal{D}$, for a monoidal subfamily $\mathcal{L}$ of 1-Lipschitz functions ${\mathrm{Lip}}_1(\mathbb{R})$, and prove its $\mathrm{\square }$-completeness and separability. We then construct a natural compactification of $(\mathcal{D}\mathrm{/}\mathcal{L},d_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}})$ by means of \emph{$\mathcal{L}$-pyramids} when $\mathcal{L}$ contains the clipping family. We further prove a complete limit formula for the observable diameter of ${\mathrm{Lip}}_1(\mathbb{R})$-pyramids, and show that applying our construction to Hanika–Schneider–Stumme’s embedding is compatible with the compactification and preserves the polynomial-time computability of the observable diameter.