Abstract

This paper investigates the proper class of all metric spaces considered up to isometry, equipped with the Gromov--Hausdorff distance. There constructed a pair of complete metric spaces, $X$ and $Y$ such that they have no metric spaces at zero distance, no optimal correspondence between $X$ and $Y$, and therefore no linear geodesics joining them, but there exists a geodesic between them of a different type. There also described everywhere dense subclass of the Gromov--Hausdorff class such that any two points at finite distance within this subclass can be connected by a linear geodesic.

Keywords: Metric space; Gromov--Hausdorff space; geodesic; distance-preserving function.

MSC: 53C23

DOI: 10.57016/MV-NSOC2660

Pages:  1--17