Metric Graph Approximations of Geodesic Spaces

Abstract

We study the question of approximating a compact geodesic metric space by metric graphs satisfying a uniform upper bound on their first Betti number. We prove that, up to a suitable multiplicative constant, Reeb graphs of distance functions to a point provide optimal approximation in the Gromov-Hausdsorff sense.

Omitted references are added

References in corpus (2)

• 1-Dimensional Intrinsic Persistence of Geodesic Spaces
• The Distortion of the Reeb Quotient Map on Riemannian Manifolds

Cited by in corpus (1)

• Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius