Borsuk's partition problem in four-dimensional $\ell_{p}$ space

TL;DR

This paper investigates Borsuk's partition problem in four-dimensional $ ext{l}_p$ spaces, establishing that all bounded sets can be partitioned into 16 smaller-diameter subsets, advancing understanding in geometric partitioning.

Contribution

It extends Borsuk's conjecture to four-dimensional $ ext{l}_p$ spaces for $1 \\leq p < 2$, proving a specific partition result for these spaces.

Findings

All bounded sets in 4D $ ext{l}_p$ spaces can be divided into 16 smaller-diameter subsets.

Analyzes the Banach-Mazur distance between the cube and $ ext{l}_p$ ball.

Generalizes Borsuk's partition problem to metric spaces.

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 $4\leq n\leq 63$. In this paper, we firstly discuss the Banach-Mazur distance between the $n$-dimensional cube and the $\ell_{p}$ ball $(1\leq p< 2)$, then we study the generalized Borsuk's partition problem in metric spaces and prove that all bounded sets $X$ in every four-dimensional $\ell_{p}$ space can be divided into $2^4$ subsets of smaller diameter.