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

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