ИСТИНА

The talk is dedicated to the proper class of all metric spaces considered up to isometry, equipped with the Gromov–Hausdorff distance. More precisely, we discuss the properties of geodesics in this Gromov–Hausdorff class. The Gromov–Hausdorff distance measures the degree of the difference between two metric spaces. This distance was introduced by Gromov in 1981 and was defined as the smallest Hausdorff distance between isometric images of the considered spaces. Later, an equivalent definition of this distance was given using correspondences. In Ivanov–Nikolaeva–Tuzhilin work an optimal correspondence between finite metric spaces was used to construct a geodesic between arbitrary com- pact metric spaces. Later on, the existence of optimal correspondence be- tween compact metric spaces was proved, and as a consequence, a geodesic between these spaces generated by the optimal correspondence was con- structed. Such geodesics are called linear ones. However, it is still unknown whether any pair of metric spaces at a finite distance from each other can be connected by some geodesic. The example of two metric spaces without linear geodesic between them is trivial, e.g. [0,1] and (0,1). Ghanaat found an example of two complete non- isometric metric spaces on zero distance, thus, they are not connected by linear geodesic. Hansen and the author found independently similar ex- amples on non- zero distance. However, in the both examples, one can choose another metric spaces on zero distance from the initial ones, such that the latter spaces are connected by a linear geodesic. Later, the author found an example of two complete metric spaces such that no linear geodesics connect them and there are no metric spaces on zero distance. However, there exists a geodesic that are not linear. In the talk another method to construct a geodesic will be presented. It is called Hausdorff realization. In the case 104two metric spaces have linear geodesic, then one can construct geodesic by means of Hausdorff realization. Nevertheless, there is an example of two metric spaces with no Hausdorff realization (Hansen). We will construct a geodesic in this case as well.