The problem of constructing geodesics in the Gromov–Hausdorff class: optimal Hausdorff realizations does not exists in general case

This work is devoted to the study of geodesics in the class of metric spaces endowed with the Gromov–Hausdorff distance. The study shows that the construction of a linear geodesic is impossible in the general case, even if we consider the Gromov – Hausdorff class factored by zero distances. Moreover, it is established that the optimal Hausdorff realization divides metric spaces at zero distance into equivalence classes with matching completions. It is also demonstrated how to construct a geodesic in Hansen's example using $0$-modifications. Nevertheless, it is shown that, in general, it is impossible to construct a geodesic using the optimal Hausdorff realization. This shows that geodesics in the class of metric spaces have an even richer structure, and the methods for constructing geodesics from the Gromov – Hausdorff space cannot be transferred to the class of metric spaces.