Convergence of metric transformed spaces

TL;DR

This paper studies how metric transformations affect the convergence of metric measure spaces, providing conditions for convergence transfer and illustrating with high-dimensional geometric examples.

Contribution

It establishes conditions under which convergence of metric measure spaces is preserved under metric transformations and applies these results to high-dimensional geometric spaces.

Findings

Spheres converge to Gaussian spaces as dimension increases.

Projective spaces converge to Hopf quotients of Gaussian spaces.

Conditions for convergence transfer between original and transformed spaces.

Abstract

We consider the metric transformation of metric measure spaces/pyramids. We clarify the conditions to obtain the convergence of the sequence of transformed spaces from that of the original sequence, and, conversely, to obtain the convergence of the original sequence from that of the transformed sequence, respectively. As an application, we prove that spheres and projective spaces with standard Riemannian distance converge to a Gaussian space and the Hopf quotient of a Gaussian space, respectively, as the dimension diverges to infinity.