The goal of the present work is to introduce new metric space techniques to study the intersection properties of families of balls. These new techniques add, on the one hand, to results due to Lindenstrauss on the extension of uniformly continuous functions and compact linear operators, and answer, on the other hand, questions raised by Aronszajn and Panitchpakdi on hyperconvex metric spaces. The present work is divided into two parts. In the first part, the proofs of our two main results on ball intersection properties in metric spaces are presented. The first main result states that for any integer n greater or equal to three, a complete and almostn-hyperconvex metric space is automatically n-hyperconvex. The second main result shows that in complete metric spaces, the property of being 4-hyperconvex is equivalent to the property of being finitely hyperconvex and that the analogues are true for externally4-hyperconvex and weakly externally4-hyperconvex subsets. This last result and the results proved later in this work unify the analysis of those three properties: hyperconvexity, external hyperconvexity and weak external hyperconvexity. In the second part of this work, we make the link with notions of convexity and bicombings and present applications. We extend local-to-global results on hyperconvex spaces to externally hyperconvex spaces as well as to weakly externally hyperconvex spaces. We conclude with applications of our results to the characterization of externally hyperconvex and weakly externally hyperconvex subsets of hyperconvex spaces.

中文翻译：

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度量空间中的球相交属性

目前工作的目标是引入新的度量空间技术来研究球族的交叉属性。这些新技术一方面增加了 Lindenstrauss 对一致连续函数和紧线性算子的扩展的结果，另一方面回答了 Aronszajn 和 Panitchpakdi 在超凸度量空间上提出的问题。目前的工作分为两个部分。在第一部分，我们展示了我们关于度量空间中球相交属性的两个主要结果的证明。第一个主要结果表明，对于任何整数n大于或等于三，一个完整且几乎n-超凸度量空间是自动的 n-超凸. 第二个主要结果表明，在完全度量空间中，存在的性质 4-超凸等价于存在的属性有限超凸并且类似物对于外部4-超凸和外部弱4-超凸子集。最后一个结果和本工作后面证明的结果统一了对这三个属性的分析：超凸性、外部超凸性和弱外部超凸性。在这项工作的第二部分，我们与凸性和双梳和目前的应用。我们将超凸空间的局部到全局结果扩展到外部超凸空间以及弱外部超凸空间。最后，我们将我们的结果应用于超凸空间的外部超凸和弱外部超凸子集的表征。