将有界集合划分为具有较小直径的集合
Divide bounded sets into sets having smaller diameters
摘要：
对于每个正整数$m$和每个实有限维Banach空间$X$，我们设置$β(X,m)$为$δ\in (0,1]$的下确界，使得每个具有直径$1$的集合$A\subset X$可以表示为直径至多为$δ$的$A$的$m$子集的并集。给出了$β(X,m)$的一些基本性质，包括在Banach-Mazur度量意义下它相对于$X$的稳定性.介绍了两种估算$β(X,m)$的方法。第一个估计$β(X,m)$使用的知识$β(Y,m)$，其中$Y$是一个Banach空间足够接近$X$。第二估计使用关于$β_X(K,m)$的信息，即$δ\in(0,1]$的下确界，使得对于$X$中的某些类凸体$K$，$K\subset X$是直径不大于$δ$乘以$K$的直径的$m$子集的并集。特别是，我们表明，$β(l_p^3,8)\leq 0.925$持有每个$p\in [1,+\infty]$通过应用第一种方法，我们证明了$β(X,8)<1$时，$X$是一个三维Banach空间满足$β_X(B_X,8)<\frac{221}{328}$，其中$B_X$是单位球的$X$，通过应用第二种方法。这些结果和方法与Borsuk问题在有限维Banach空间中的推广以及C。Zong的Borsuk猜想计算机程序。
英文摘要：
For each positive integer $m$ and each real finite dimensional Banach space $X$, we set $β(X,m)$ to be the infimum of $δ\in (0,1]$ such that each set $A\subset X$ having diameter $1$ can be represented as the union of $m$ subsets of $A$ whose diameters are at most $δ$. Elementary properties of $β(X,m)$, including its stability with respect to $X$ in the sense of Banach-Mazur metric, are presented. Two methods for estimating $β(X,m)$ are introduced. The first one estimates $β(X,m)$ using the knowledge of $β(Y,m)$, where $Y$ is a Banach space sufficiently close to $X$. The second estimation uses the information about $β_X(K,m)$, the infimum of $δ\in(0,1]$ such that $K\subset X$ is the union of $m$ subsets having diameters not greater than $δ$ times the diameter of $K$, for certain classes of convex bodies $K$ in $X$. In particular, we show that $β(l_p^3,8)\leq 0.925$ holds for each $p\in [1,+\infty]$ by applying the first method, and we proved that $β(X,8)<1$ whenever $X$ is a three-dimensional Banach space satisfying $β_X(B_X,8)<\frac{221}{328}$, where $B_X$ is the unit ball of $X$, by applying the second method. These results and methods are closely related to the extension of Borsuk's problem in finite dimensional Banach spaces and to C. Zong's computer program for Borsuk's conjecture.
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