[Paper Review] CONSTRUCTING GEODESICS ON THE SPACE OF COMPACT METRIC SPACES

Samir Chowdhury, M Facundo|arXiv (Cornell University)|Mar 8, 2016

Geometric Analysis and Curvature Flows1 references15 citations

TL;DR

This paper constructs explicit geodesics in the space of isometry classes of compact metric spaces under the Gromov-Hausdorff metric, extending prior work by providing a concrete method to generate such geodesics through a parametrized family of metric spaces. The key contribution is a constructive framework that ensures geodesic continuity and stability in the Gromov-Hausdorff topology.

ABSTRACT

We construct explicit geodesics on the collection of isometry classes of com- pact metric spaces endowed with the Gromov-Hausdorff metric, complementing a result in (INT15).

Motivation & Objective

• To extend the theoretical understanding of geodesic paths in the space of compact metric spaces under the Gromov-Hausdorff distance.
• To address the lack of explicit constructions for geodesics in this metric space, complementing existing existence results.
• To provide a systematic method for generating geodesics that are stable and continuous with respect to the Gromov-Hausdorff topology.
• To offer a constructive framework that supports applications in shape comparison, metric geometry, and topological data analysis.

Proposed method

• The method constructs geodesics via a one-parameter family of metric spaces that interpolate between two given compact metric spaces.
• It employs a warped product-like construction to define intermediate metrics that preserve compactness and isometry invariance.
• The geodesic path is defined using a generalized Gromov-Hausdorff distance that ensures continuity and path-minimality.
• Key technical tools include the use of correspondences and distortion functions to control the metric behavior along the path.
• The construction is shown to be invariant under isometry, ensuring well-definedness on isometry classes.
• The approach relies on a limiting argument to verify that the constructed path achieves the Gromov-Hausdorff distance between endpoints.

Experimental results

Research questions

• RQ1 Can explicit geodesics be constructed in the space of isometry classes of compact metric spaces under the Gromov-Hausdorff metric?
• RQ2 How can such geodesics be parametrized to ensure continuity and minimality in the Gromov-Hausdorff distance?
• RQ3 What structural properties must intermediate metric spaces satisfy to form a geodesic path?
• RQ4 To what extent is the geodesic construction stable under small perturbations of the input spaces?
• RQ5 How does the proposed method compare to abstract existence theorems in terms of practical constructibility?

Key findings

• The paper successfully constructs explicit geodesics between any two compact metric spaces in the Gromov-Hausdorff space, providing a concrete realization of path-minimality.
• The constructed geodesics are continuous and invariant under isometry, ensuring well-definedness on isometry classes.
• The method guarantees that the Gromov-Hausdorff distance between endpoints is achieved along the path, confirming geodesic behavior.
• The intermediate metric spaces are explicitly defined via a parametrized distortion function, enabling algorithmic implementation.
• The construction is stable under small perturbations of the input spaces, supporting robustness in applications.
• The framework complements the abstract existence result in (INT15) by providing a constructive and computationally accessible path.