Classification of Six-Point Metrics

Classification of finite metric spaces and combinatorics of convex polytopes

We describe the canonical correspondence between set of all finite metric spaces and set of special symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those polytopes.

色度度量对之间的 Gromov-Hausdorff 距离与 six-pack 的稳定性

Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack

本文提出色度度量对间推广的 Gromov-Hausdorff 距离，并证明 six-pack 与 Čech 持续图相对该距离的稳定性。

Classification of Finite Subspaces of Metric Space Instead of Constraints on Metrics

A new method of metric space investigation, based on classification of its finite subspaces, is suggested. It admits to derive information on metric space properties which is encoded in metric. The method describes geometry in terms of only metric. It admits to remove constraints imposed usually on metric (the triangle axiom and nonnegativity of the squared metric), and to use the metric space for description of the space-time and other geometries with indefinite metric. Describing space-time and using this method, one can explain quantum effects as geometric effects, i.e. as space-time proper…

Six mathematical gems from the history of Distance Geometry

This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron's formula, Cauchy's theorem on the rigidity of polyhedra, Cayley's generalization of Heron's formula to higher dimensions, Menger's characterization of abstract semi-metric spaces, a result of Goedel on metric spaces on the sphere, and Schoenberg's equivalence of distance and positive semidefinite matrices, which is at the basis of Multidimensional Scaling.

Coarse classification of abelian groups and amenable shift-homogeneous metric spaces

In this paper we classify countable locally finite-by-abelian groups up to coarse isomorphism. This classification is derived from a coarse classification of amenable shift-homogeneous metric spaces.

Classification of metric spaces with infinite asymptotic dimension

We introduce a geometric property complementary-finite asymptotic dimension (coas- dim). Similar with asymptotic dimension, we prove the corresponding coarse invariant theorem, union theorem and Hurewicz-type theorem.