On the metric geometry of the space of compact balls and the shooting property for length spaces

TL;DR

This paper investigates the geometric structure of the space of compact balls in length spaces, establishing an explicit isometry under the shooting property that relates it to a half-space with a taxicab metric.

Contribution

It introduces the shooting property for length spaces and provides an explicit isometry between the space of compact balls and a half-space with a taxicab metric.

Findings

The space of compact balls can be isometrically embedded into a half-space with a taxicab metric.

The shooting property characterizes when this isometry exists.

Explicit construction of the isometry for spaces with the shooting property.

Abstract

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.