A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces

By Romain Abraham, Jean-François Delmas, Patrick Hoscheit. 2012

131 citations recorded by the source corpus.

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a boundedly finite measure. We prove that this space with the extended Gromov-Hausdorff-Prokhorov metric is a Polish space. This generalization is needed to define Levy trees, which are (possibly unbounded) random real trees endowed with a boundedly finite measure.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1c5ea8b7759a93a63f303d28409b6d58ae17c0a9

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Convergence of pointed non‐compact metric measure spaces and stability of Ricci curvature bounds and heat flows (2013, background): “Let us finally mention that (B′) gives the same convergence as the Gromov-HausdorffProkhorov metric studied in [1] in case of locally compact length spaces endowed with locally finite measures; here the authors prove that such a metric induces a structure of Polish space on the equivalence classes o”
The local weak limit of the minimum spanning tree of the complete graph (2013, background): “Interested readers can find proofs of the various assertions regarding convergence of metric spaces in, for example, [1, 30, 55].”