Metrization of space of points families in Rn and adjoining questions

In the work we introduced the concept of a family of points in R𝑛 and metrization of space of points families. Under the family we understood the points numbered set of points in R𝑛. In the interpretation of points in space as the nodes of a grid (lattice) introduced in this paper, the concept of distance can be used as a kind of measure of the differences between the test grid of reference or in some critical sense. Moreover, this measure of the difference can be determined through measure differences corresponding to the grid elements - for example, in the case of tetrahedral mesh - for its individual tetrahedrons adjacent, for couples tetrahedra. A family of points (𝑘-point family) is a function : {1,..., 𝑘} → R𝑛. We define the distance (𝐹,𝐺) between the families and as the logarithm of some expression that contains the Euclidean distance |𝐹(𝑖)𝐹(𝑗)|, |𝐺(𝑖)𝐺(𝑗)|. Distance is invariant relatively orthogonal mapping: (𝑂 ∘ 𝐹,𝐺) = (𝐹,𝐺) for any orthogonal mapping : R𝑛 → R𝑛. We give an estimate of the distance that moves the family under the action of quasi-isometric mapping 𝑓: (𝐹, ∘𝐹) ≤ ≤ log 𝑙, where is minimum distortion mapping 𝑓, is maximum distortion mapping 𝑓. Next, we prove the following sufficient sign of preservation any properties of families of points at quasi-isometric mapping: Тheorem 2. Let is 𝑘-point family in R𝑛, : 𝐹(𝐼) → R𝑛 is quasi-isometric mapping; - a set of 𝑘-point families. If 𝐹 ̸∈ and log 𝑙