. We show that the problem whether a given finite metric space can be embedded into m-dimensional rectilinear space can be reformulated in terms of the Gromov- Hausdorff distance between some special finite metric spaces.

The paper is devoted to geometrical investigation of Gromov–Hausdorff distance on the classes of all metric spaces and of all bounded metric spaces. The main attention is paid to path connectivity questions. The path connected components of the Gromov–Hausdorff class of all metric spaces are described, and the path connectivity of spheres is proved in several particular cases.

An implementation of a “rectilinear” geodesic lying in the Gromov–Hausdorff space is constructed in the form of the shortest geodesic with respect to the Hausdorff distance in some ambient metric space.

The aim of this paper is to demonstrate relations between Gromov–Hausdorff distance properties and the Borsuk Conjecture. The Borsuk number of a given bounded metric space X is the infimum of cardinal numbers n such that X can be partitioned into n smaller parts (in the sense of diameter). An exact formula for the Gromov–Hausdorff distance between bounded metric spaces is obtained under the assumptions that the diameter and the cardinality of one space is less than the diameter and the Borsuk number of the other one, respectively, see Theorem 4.1. Using Bacon equivalence results between Lusternik–Schnirelmann and Borsuk Problems several corollaries are obtained.

The purpose of this article is to demonstrate the connection between the properties of the Gromov–Hausdorff distance and the Borsuk conjecture. The Borsuk number of a given bounded metric space X is the infimum of cardinal numbers n such that X can be partitioned into n smaller parts (in the sense of diameter). An exact formula for the Gromov–Hausdorff distance between bounded metric spaces is obtained under the assumption that the diameter and cardinality of one space are less than the diameter and Borsuk number of another, respectively. Using the results of Bacon's equivalence between the Lusternik–Schnirelmann and Borsuk problems, several corollaries are obtained.

В работе показано, что любое ограниченное метрическое пространство изометрично вкладывается в метрический класс Громова-Хаусдорфа $\operatorname{\mathcal{GH}}$. Этот результат является следствием полученного в работе описания локальной геометрии $\operatorname{\mathcal{GH}}$ в достаточно малой окрестности метрического пространства общего положения, которое представляет самостоятельный интерес. Использована техника оптимальных соответствий и их искажений. Библиография: 22 названия.

Кузин Сергей Сергеевич -ведущий инженер, ООО

The Fermat-Steiner problem consists in finding all points in a metric space at which the sum of the distances to fixed points of attains its minimum value. This problem is studied in the metric space of all nonempty compact subsets of the Euclidean space , and the are pairwise disjoint finite sets in . The set of solutions of this problem (which are called Steiner compact sets) falls into different classes in accordance with the distances to the . Each class contains an inclusion-greatest element and inclusion-minimal elements (a maximal Steiner compact set and minimal Steiner compact sets, respectively). We find a necessary and sufficient condition for a compact set to be a minimal Steiner compact set in a given class, provide an algorithm for constructing such compact sets and find a sharp estimate for their cardinalities. We also put forward a number of geometric properties of minimal and maximal compact sets. The results obtained can significantly facilitate the solution of specific problems, which is demonstrated by the well-known example of a symmetric set , for which all Steiner compact sets are asymmetric. The analysis of this case is significantly simplified due to the technique developed. Bibliography 16 titles.

Urolithiasis is currently one of the most urgent problems in the world. Every eleventh worldwide inhabitant suffers from this disease. Previously, the only way to get rid of kidney stones and the urinary tract was open surgery, which was characterized by high trauma. Over the past decades, the development of technologies has made a significant contribution to the development of new methods of urolithiasis treatment. One of these methods is extracorporeal shock wave lithotripsy (ESWL). The first lithotripter Dornier HM-1 was produced in 1980. Subsequent models have got many changes, both in terms of ergonomics and power. The researchers noticed that the efficiency of stone crushing in the Dornier HM-1 lithotripter was higher than in newer models since the lower power provided the less intensive formation of cavitation bubbles that prevent the effective transit of subsequent waves through the stone. Nowadays, a new method of remote stone crushing is being developed based on low-amplitude high-frequency technology combined with ultrasonic propulsion, which is the main difference from traditional shock-wave lithotripters. The new technology of stone crushing is called «burst wave lithotripsy» (BWL). Currently, the data have been obtained that this method is more effective in terms of crushing quality and less traumatic.

The present paper is devoted to studying of minimal parametric fillings of finite metric spaces (a version of optimal connection problem) by methods of Linear Programming. The estimate on the multiplicity of multi-tours appearing in the formula of weight of minimal fillings is improved, an alternative proof of this formula is obtained, and also explicit formulas for finite spaces consisting of 5 and 6 points are derived.

In this paper geometry of Gromov-Hausdorff distance on the class of all metric spaces considered up to an isometry is investigated. For this class continuous curves and their lengths are defined, and it is shown that the Gromov-Hausdorff distance is intrinsic. Besides, metric segments are considered, i.e., the classes of points lying between two given ones, and an extension problem of such segments beyond their end-points is considered.

The behavior of extreme networks under deformations of their boundary sets is investigated. It is shown that analyticity of a deformation of the boundary set guarantees preservation of the network type for minimal spanning trees, minimal fillings, and so-called stable shortest trees in the Euclidean space.

Improvement of patients` knowledge about their disease is an important part of management of chronic diseases. One of the effective methods to increase the level of medical education of the population is to hold regular meetings with patients within the framework of «schools for patients». Urolithiasis is one of the most common urological diseases (3.5‒9.6% of the population). In addition to its high prevalence, Urolithiasis has a high recurrence rate (50‒75% over 5‒10 years). The most common prostate diseases are prostatitis, benign hyperplasia, and cancer. These diseases have a chronic course and require а long-term observation. Treatment of urolithiasis and prostatic diseases is one of the priority fields for the urology department of the Pavlov First St. Petersburg State Medical University. In addition to surgical treatment, the specialists of urology department take measures aimed at prevention of recurrence and complications of these diseases. Schools for patients with urolithiasis and prostate diseases are held every 3 months. The topics for discussion are chosen by the patients themselves. In addition to reports, patients undergo ultrasound of the kidneys, bladder and prostate, as well as consultations. Participation in schools of patients creates a conscious attitude to their disease, increases adherence to treatment and improves the prognosis of the disease. The distribution of such programs corresponds to the modern concept of preventive medicine and increases the efficiency of providing medical care to the population.

The Relativistic one dimensional Coulomb problem was studied by means of the Path Integral Monte Carlo method. Relativistic and non-relativistic regimes of this problem were investigated. The relativistic regime appears at small masses of the particle and (or) at strong coupling. Critical coupling at which the bound state disappears was founded. This law is valid for any finite regularization of the Coulomb potential and potentially can be tested experimentally.

The present paper is devoted to investigation of the isometry group of the Gromov-Hausdorff space, i.e., the metric space of compact metric spaces considered up to an isometry and endowed with the Gromov-Hausdorff metric. The main goal is to present a proof of the following theorem by George Lowther (2015): The isometry group of the Gromov-Hausdorff space is trivial. Unfortunately, the author himself has not publish an accurate text for 2 years passed from the publication of draft (that is full of excellent ideas mixed with unproved and wrong statements) in the https://mathoverflow.net/ blog (see the exact reference in he bibliography).

The Steiner problem is considered in the Gromov-Hausdorff space, i.e., in the space of compact metric spaces (considered up to isometry) endowed with the Gromov-Hausdorff distance. Since this space is not boundedly compact, the problem of the existence of a shortest network in this space is open. It is shown that each finite family of finite metric spaces can be connected by a shortest network. Moreover, it turns out that in this case there exists a shortest tree all of whose vertices are finite metric spaces. An estimate for the number of points in these metric spaces is obtained. As an example, the case of three-point metric spaces is considered. It is also shown that the Gromov-Hausdorff space does not realize minimal fillings; i.e., shortest trees in this space need not be minimal fillings of their boundaries.