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

TL;DR

This paper investigates geodesic extendibility in complete, locally compact length spaces and explores the structure of the space of compact balls with the Hausdorff metric, revealing isometries and applications to isometry groups.

Contribution

It provides an explicit isometry between the space of compact balls and a half-space with a taxicab metric, and establishes a group isometry for Hadamard spaces.

Findings

Explicit isometry between $(\Sigma (X),d_H)$ and a half-space with taxicab metric

Group isometry between isometry groups of $X$ and $\Sigma (X)$ for Hadamard spaces

Insights into geodesic extendibility in length spaces

Abstract

In this work we study the issue of geodesic extendibility on complete and locally compact metric length spaces. We focus on the geometric structure of the space $(\Sigma (X),d_H)$ of compact balls endowed with the Hausdorff distance and 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. 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.