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Hopf quotients of the infinite-dimensional Gaussian pyramid
Takashi Shioya, Shigeaki Yokota
We study the infinite-dimensional Gaussian pyramid and its quotients by the global sign flip and the -Hopf action. We resolve affirmatively a long-standing problem posed by T…
Compact Screens and Pyramidal Compactification of Geometric Data Sets
Shigeaki Yokota
We introduce an observable distance that compares realvalued features through a fixed bounded coordinate. The coordinate retains the distinction between finite feature values while…
Funk Beta Balls: Exact Potentials and a Three-Phase Pyramid Diagram
Shigeaki Yokota
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 pyr…
Extensions of One-Sided Box Geometry and Pyramid Invariants to gd-Sets and qm-Spaces
Shigeaki Yokota
Domination of gd-sets, the relation recording which function family approximates which, is closed under box convergence. More generally, approximate 1-Lipschitz maps on asymptotica…
Radial Hyperbolic Measures: Shell Geometry, Pyramid Limits, and Gaussian Phase Transitions
Shigeaki Yokota
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 sepa…
Pyramidal Compactification of Asymmetric Metric Measure Spaces via Adjoint Transport
Shigeaki Yokota
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 sup…
Poincaré Beta Balls: Radial Laws, Hyperbolic Shell Metric Transforms, and Critical-Limit Regimes
Shigeaki Yokota
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 sh…
Geometry of Geometric Data Set II: Pyramid
Shigeaki Yokota
The observable distance based on measure concentration and the box distance based on collapsing theory are extended to geometric data sets introduced by…
Geometry of Geometric Data Set I
Shigeaki Yokota
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…
A Complete Proof of the Limit Formula for Observable Diameter
Shigeaki Yokota
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 the…