Banach-Mazur Distance from ${\ell }_p^3$ to ${\ell }_{\infty }^3$

Abstract

The maximum of the Banach-Mazur distance $d_{BM}^M(X,{\ell }_{\infty }^n)$, where $X$ ranges over the set of all $n$-dimensional real Banach spaces, is difficult to compute. In fact, it is already not easy to get the maximum of $d_{BM}^M({\ell }_p^n,{\ell }_{\infty }^n)$ for all $p\in [1,\infty ]$. We prove that $d_{BM}^M({\ell }_p^3,{\ell }_{\infty }^3)\le 9/5,\forall p\in [1,\infty ]$. As an application, the following result related to Borsuk's partition problem in Banach spaces is obtained: any subset $A$ of ${\ell }_p^3$ having diameter $1$ is the union of $8$ subsets of $A$ whose diameters are at most $0.9$.