论文标题
Banach-Mazur距离$ \ ell_p^3 $到$ \ ell_ \ infty^3 $的距离
Banach-Mazur Distance from $\ell_p^3$ to $\ell_\infty^3$
论文作者
论文摘要
Banach-Mazur距离的最大值$ d_ {bm}^m（x，\ ell_ \ infty^n）$，其中$ x $在所有$ n $ dimensional的真实Banach空间的集合上都很难计算。实际上，对于所有$ d_ {bm}^m的最大值（\ ell_p^n，\ ell_ \ infty^n）$的最大值对于[1，\ infty] $中的全部$ p \而言。我们证明$ d_ {bm}^m（\ ell_p^3，\ ell_ \ infty^3）\ leq 9/5，〜\ forall p \ in [1，\ infty] $。作为一个应用程序，获得了与Borsuk在Banach空间中的分区问题有关的以下结果：任何$ \ ell_p^3 $具有直径$ 1 $的子集$ a $ a $ a $ \ ell_p^3 $是$ 8 $ $ 8 $ a的$ a $ a $的联合，其直径最高为$ 0.9 $。
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 $d_{BM}^M(\ell_p^3,\ell_\infty^3)\leq 9/5,~\forall p\in[1,\infty]$. 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 $0.9$.
