Borsuk's partition problem in four-dimensional ${\ell }_p$ space

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 $4\le n\le 63$. In this paper, we firstly discuss the Banach-Mazur distance between the $n$-dimensional cube and the ${\ell }_p$ ball $(1\le p<2)$, 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.