The paper elucidates the relationship between the density of a Banach space and possible sizes of well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $X$, the unit sphere $S_X$ always contains an uncountable $(1+)$-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established that happen to be sharp for the class of WLD spaces, as witnessed by a renorming of $c_0(\omega_1)$ without uncountable Auerbach systems. The following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically $(1+\varepsilon)$-separated subset of any regular cardinality not exceeding the density of $X$; should the space $X$ be super-reflexive, the unit sphere of $X$ contains such a subset of cardinality equal to the density of $X$. The said problem is studied for other classes of spaces too, including the RNP spaces or strictly convex ones.

中文翻译：

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Banach空间中的分离集和Auerbach系统

该论文阐明了 Banach 空间的密度与其单位球体的分离良好的子集的可能大小之间的关系。例如，证明对于足够大的空间$X$，单位球$S_X$ 总是包含不可数的$(1+)$-分隔的子集。为了实现这一点，建立了关于大型 Auerbach 系统存在的新结果，这些结果恰好对 WLD 空间类来说是尖锐的，正如 $c_0(\omega_1)$ 没有不可数的 Auerbach 系统的重新归一所证明的那样。分别建立了自反空间和超自反空间类别的以下最优结果：无限维自反空间的单位球面包含对称的 $(1+\varepsilon)$ 分隔的任何规则基数不超过$X$的密度；如果空间 $X$ 是超自反的，则 $X$ 的单位球面包含这样一个基数子集，等于 $X$ 的密度。上述问题也适用于其他类别的空间，包括 RNP 空间或严格凸空间。