On geodesic extendibility and the space of compact balls of length spaces

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