Surjective Mappings in the Hyers--Ulam Theorem and the Gromov--Hausdorff Distance

TL;DR

This paper explores conditions under which approximate isometries between certain homogeneous metric spaces can be approximated by true isometries, linking the Hyers--Ulam stability to the Gromov--Hausdorff distance and Banach space isometries.

Contribution

It establishes a new connection between $ ext{d}$-isometries and bijective isometries in cardinality homogeneous metric spaces, extending the Hyers--Ulam theorem.

Findings

Existence of a bijective $(d+2\delta)$-isometry under $ ext{ extdelta}$-surjective $d$-isometries.

Reduction of the Dilworth--Tabor theorem to the Gevirtz--Omladič--Šemrl theorem.

Implications for the isometry problem in Banach spaces.

Abstract

A topological space is said to be cardinality homogeneous if every nonempty open subset has the same cardinality as the space itself. Let $X$ and $Y$ be cardinality homogeneous metric spaces of the same cardinality. If there exists a $\delta$-surjective $d$-isometry between such equicardinal cardinality homogeneous metric spaces $X$ and $Y$, then there exists a bijective $(d+2\delta)$-isometry between $X$ and $Y$. This result allows us to reduce the Dilworth--Tabor theorem to the Gevirtz--Omladi\v{c}--\v{S}emrl theorem on approximation by isometries and, in particular, to questions concerning the isometry of Banach spaces.