arXiv subjects
Publications and source records attributed to Shigeaki Yokota.
We study the infinite-dimensional Gaussian pyramid and its quotients by the global sign flip and the $U(1)$-Hopf action. We resolve affirmatively a long-standing problem posed by Tomohiro Fukaya around 2014: these three limiting geometries are pairwise non-similar, meaning that no positive rescaling makes any two of them coincide.
We introduce an observable distance that compares realvalued features through a fixed bounded coordinate. The coordinate retains the distinction between finite feature values while compressing their independent escape to infinity, the source of nonseparability for the classical observable distance on all geometric data sets. The new distance makes the full class separable and geodesic: every pair is joined by a constant-speed path, and on metric measure spaces the induced topology agrees with the concentration topology. From the same coordinate we construct compact screens, whose features take values in one fixed compact interval. The screened class is Polish and geodesic for the Box distance. Organizing its finite-feature quotients by the feature order yields a compact pyramid space. Pyramids generated by single compact screens form a dense subspace, so this pyramid space compactifies the original class after passage to compact screens. Convergence is detected through the Box-Hausdorff behavior of every finite measurement layer. Finally, taking sum-metric products with a common metric measure factor is nonexpansive for the new distance. More precisely, the comparison determined by a prescribed coupling of the original spaces and the diagonal coupling of the common factor is preserved, whereas optimization over all product couplings yields the nonexpansive inequality.
In a high-dimensional Funk ball, the endpoint potential makes the forward and reverse distances behave differently, while symmetrization erases the distinction that survives in pyramid limits. For Euclidean beta-type radial measures, we determine the natural-scale limits in all three parameter phases. The divergent phase yields directed Gaussian pyramids indexed by the balance between dimension and radial concentration. A positive limiting parameter yields a Gaussian--chi Funk horocone pyramid, built from Gaussian bases, chi-distributed heights, and a directed endpoint-potential increment, while a vanishing parameter yields the universal directed star pyramid. In the last phase, the dimension-dependent logarithmic contributions cancel exactly, so the conclusion requires no further rate condition.
In high-dimensional hyperbolic space, concentration of a radial measure near a shell need not determine the pyramid limit: angular concentration and hyperbolic expansion alter separation. An effective radius is the scale on which positive-mass sets actually separate, which the raw radius need not give. With vanishing rescaling and radial fluctuations, radius convergence gives weak convergence to the pyramid of all metric measure spaces with the resulting diameter bound. At modal radial Gibbs shells, exponential decay of intrinsic shell curvature and dimension-normalized tangential Bakry-Émery Ricci curvature recovers the radius. The Gaussian intrinsic to hyperbolic volume and that obtained by wrapping a Euclidean Gaussian have different critical orders. They are Lévy below those orders, infinitely dissipate above them, and at criticality converge to the corresponding diameter-bounded pyramids.
Domination of gd-sets, the relation recording which function family approximates which, is closed under box convergence. More generally, approximate 1-Lipschitz maps on asymptotically full-measure subsets, whose restricted measures converge to the target measure after pushforward, place the target representation in every subsequential weak limit of the source pyramids. For weakly convergent pyramids, bounded joint distributions of finite ordered tuples of observables recover both observable diameter, after rightward perturbation of its mass parameter, and the largest common forward gap among several positive-mass sets, after common leftward perturbation of their mass parameters. The lower- and upper-limit formulas agree as the perturbations vanish. Without closure assumptions on a gd-set's function family, we compare observable diameter, the nonnegative part of forward separation, and upper and lower median-tail masses. All observables become uniformly close in measure to suitable constants exactly when both median-tail masses vanish at every positive radius.
A quasi-metric measure space (qm-space) is a set with a directed distance whose symmetrization is a complete separable metric, together with a Borel probability measure of full support. Motivated by the problem of determining the pyramid limits of beta measures on forward Funk balls, we construct a compact metric space of pyramids of qm-spaces. The associated-pyramid map from the concentration-distance space of qm-spaces into this compact space is a $1$-Lipschitz topological embedding with dense image. As a secondary result, we prove that the box-distance space of qm-spaces is complete and separable.
On a high-dimensional Poincaré ball, a Euclidean beta-type radial measure concentrates near a sphere, but hyperbolic distance amplifies the surviving radial spread, so the usual shell reduction loses part of the limiting metric. In each regime of the balance between this radial width and amplified angular separation, we determine the weak pyramid limit of the spaces rescaled at the order at which the transition between concentration and dissipation occurs. The four possibilities are a pyramid generated by finite star trees, the pyramid of spaces of diameter at most one, metric transforms of the Gaussian pyramid, and the Gaussian pyramid. Each star tree has branches from a common center, a shifted exponential distribution along them, and paths between different branches through the center. We also give a sharp criterion for convergence to the diameter-at-most-one 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.
Hanika, Schneider, and Stumme introduced geometric data set as a generalization of metric measure space for the computation of the observable diameter, and extended the observable distance between metric measure spaces to that between geometric data sets. In this paper, we begin by proving the non-separability of the observable distance between geometric data sets. We then extend the box distance between mm-spaces to that between geometric data sets and prove its completeness and non-separability.
Ozawa and Shioya proposed the limit formula for observable diameters of pyramids under weak convergence. However, we find a constructive counterexample to an inequality used in their proof. In this paper, we correct the inequality and verify the limit formula.