WIAS Preprint No. 2458, (2017)

Geometric properties of cones with applications on the Hellinger--Kantorovich space, and a new distance on the space of probability measures


Authors

  • Laschos, Vaios
    ORCID: 0000-0001-8721-5335
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 51Fxx

Keywords

  • Geometry on cones, local angle condition, K-semiconcavity, Hellinger-Kantorovich, Spherical Hellinger-Kantorovich

DOI

10.20347/WIAS.PREPRINT.2458

Abstract

By studying general geometric properties of cone spaces, we prove the existence of a distance on the space of Probability measures that turns the Hellinger--Kantorovich space into a cone space over the space of probabilities measures. Here we exploit a natural two-parameter scaling property of the Hellinger-Kantorovich distance. For the new space, we obtain a full characterization of the geodesics. We also provide new geometric properties for the original space, including a two-parameter rescaling and reparametrization of the geodesics, local-angle condition and some partial K-semiconcavity of the squared distance, that it will be used in a future paper to prove existence of gradient flows.

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