In this work we study the issue of geodesic extendibility on complete and locally compact metric length spaces. We focus on the geometric structure of the space \((\varSigma (X),d_H)\) of compact balls endowed with the Hausdorff distance and give an explicit isometry between \((\varSigma (X),d_H)\) and the closed half-space \( X\times {\mathbb {R}}_{\ge 0}\) endowed with a taxicab metric. Among the applications we establish a group isometry between \(\text {Isom}(X,d)\) and \(\text {Isom}(\varSigma (X),d_H)\) when (X, d) is a Hadamard space.

中文翻译：

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关于测地线的可扩展性和长度空间的紧致球的空间

在这项工作中，我们研究了完整和局部紧凑的公制长度空间上的测地线可扩展性问题。我们关注具有Hausdorff距离的紧致球的空间\（（\ varSigma（X），d_H）\）的几何结构，并给出\（（\ varSigma（X），d_H）\）和封闭的半空间\（X \次{\ mathbb {R}} _ {\ ge 0} \）拥有计程车度量。在应用程序中，当（X，  d）是a时，我们在\（\ text {Isom}（X，d）\）和\（\ text {Isom}（\ varSigma（X），d_H）\）之间建立群等距。哈达玛空间。