Solution to Generalized Borsuk Problem in Terms of the Gromov-Hausdorff Distances to Simplexes

Alexander Ivanov; Alexei Tuzhilin

Solution to Generalized Borsuk Problem in Terms of the Gromov-Hausdorff Distances to Simplexes

Metric Geometry 2019-06-26 v1 Functional Analysis

Authors: Alexander Ivanov , Alexei Tuzhilin

Abstract

In the present paper the following Generalized Borsuk Problem is studied: Can a given bounded metric space $X$ be partitioned into a given number $m$ (probably an infinite one) of subsets, each of which has a smaller diameter than $X$ ? We give a complete answer to this question in terms of the Gromov-Hausdorff distance from $X$ to a simplex of cardinality $m$ and having a diameter less than $X$ . Here a simplex is a metric space, all whose non-zero distances are the same.

Keywords

generalized borsuk problem gromov-hausdorff distance metric space partition simplex diameter

Cite

@article{arxiv.1906.10574,
  title  = {Solution to Generalized Borsuk Problem in Terms of the Gromov-Hausdorff Distances to Simplexes},
  author = {Alexander Ivanov and Alexei Tuzhilin},
  journal= {arXiv preprint arXiv:1906.10574},
  year   = {2019}
}

Solution to Generalized Borsuk Problem in Terms of the Gromov-Hausdorff Distances to Simplexes

Abstract

Keywords

Cite

Related papers