Geometry of Linear and Nonlinear Geodesics in the Proper Gromov--Hausdorff Class

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.