Borsuk's Problem in Metric Spaces

2022/10/12 by Jun Wang, Fei Xue, Wang, Jun +3
Mathematics · #52C17 #Advanced Topology and Set Theory #FOS: Mathematics #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2210.06264

openalex publication_date 2022/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1933, K. Borsuk proposed the following problem: Can every bounded set in 𝔼ⁿ be divided into n+1 subsets of smaller diameters? In 1965, V. G. Boltyanski and I. T. Gohberg made the following conjecture: Every bounded set in an n-dimensional metric space can be divided into 2ⁿ subsets of smaller diameters. In this paper, we prove the following result: Every bounded set in an n-dimensional metric space can be divided into 2ⁿ((n+1)log (n+1)+(n+1)log log (n+1)+5n+5) subsets of smaller diameters.

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