  In this work we study the geodesic structure of the space $\Sigma (X)$ of
compact balls of a complete and locally compact metric length space endowed
with the Hausdorff distance $d_H$. In particular, we focus on a geometric
condition (referred to as the shooting property) that enables us to give an
explicit isometry between $(\Sigma (X),d_H)$ and the closed half-space $
X\times \mathbb{R}_{\ge 0}$ endowed with a taxicab metric.
