Geometry of Geometric Data Set II: Pyramid

The observable distance $d_{\mathrm{conc}}$ based on measure concentration and the box distance $\Box$ 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, $d_{\mathrm{conc}}$ is non-separable and $\Box$ is complete and non-separable. We introduce the class $\mathcal{D}/\mathcal{L}$ of $\mathcal{L}$-compact geometric data sets in $\mathcal{D}$, for a monoidal subfamily $\mathcal{L}$ of 1-Lipschitz functions $\operatorname{Lip}_1(\mathbb{R})$, and prove its $\Box$-completeness and separability. We then construct a natural compactification of $(\mathcal{D}/\mathcal{L}, d_{\mathrm{conc}})$ 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 $\operatorname{Lip}_1(\mathbb{R})$-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.