WIAS Preprint No. 2207, (2016)

Optimal entropy-transport problems and a new Hellinger--Kantorovich distance between positive measures


  • Liero, Matthias
    ORCID: 0000-0002-0963-2915
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Savaré, Giuseppe
    ORCID: 0000-0002-0104-4158

2010 Mathematics Subject Classification

  • 28A33 54E35 49Q20 49J35 49J40 49K35 46G99


  • Entropy-transport problem, Hellinger-Kantorovich distance, relative entropy, optimality conditions, cone over metric space




We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.

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