CAT（0）空间中的紧集及其凸包的闭包
Compact sets and the closure of their convex hulls in CAT(0) spaces
摘要：
研究完备CAT（0）空间中紧集的凸包的闭包。首先给出了局部紧CAT（0）空间的紧集及其凸包的闭包的特征结果，其中CAT（0）空间是正则的或满足测地扩张性质.后来受到启发的几何解释Carathéodory定理，我们介绍了操作线程为一个给定的集合。我们表明，线程表现出一定的单调性属性的交集和并集。此外，线程保持紧凑性。其次，从穿线与任意等距映射的交换性出发，证明了在平坦完备CAT（0）空间中，紧集的凸包的闭包是紧的。我们应用我们的理论的Fréchet平均值的一个有限的点集的可计算性，并表明，它是在最多一个有限数量的步骤，每当潜在的空间是有限型的建设。
英文摘要：
We study the closure of the convex hull of a compact set in a complete CAT(0) space. First we give characterization results in terms of compact sets and the closure of their convex hulls for locally compact CAT(0) spaces that are either regular or satisfy the geodesic extension property. Later inspired by a geometric interpretation of Carathéodory's Theorem we introduce the operation of threading for a given set. We show that threading exhibits certain monotonicity properties with respect to intersection and union of sets. Moreover threading preserves compactness. Next from the commutativity of threading with any isometry mapping we prove that in a flat complete CAT(0) space the closure of the convex hull of a compact set is compact. We apply our theory to the computability of the Fréchet mean of a finite set of points and show that it is constructible in at most a finite number of steps, whenever the underlying space is of finite type.
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