Borsuk's partition problem in four-dimensional ℓ_p space

2022/06/30 by Wang, Jun, Xue, Fei
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2206.15277

Abstract

In 1933, Borsuk made a conjecture that every n-dimensional bounded set can be divided into n+1 subsets of smaller diameter. Up to now, the problem is still open for 4≤ n≤ 63. In this paper, we firstly discuss the Banach-Mazur distance between the n-dimensional cube and the ℓ_p ball (1≤ p< 2), then we study the generalized Borsuk's partition problem in metric spaces and prove that all bounded sets X in every four-dimensional ℓ_p space can be divided into 2⁴ subsets of smaller diameter.

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