On the metric geometry of the space of compact balls and the shooting property for length spaces
Waldemar Barrera, Luis Montes de Oca, Didier A. Solis

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 of compact balls of a complete and locally compact metric length space endowed with the Hausdorff distance . In particular, we focus on a geometric condition (referred to as the shooting property) that enables us to give an explicit isometry between and the closed half-space endowed with a taxicab metric.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric Analysis and Curvature Flows · Point processes and geometric inequalities · Advanced Differential Geometry Research
