Metric Properties of Homogeneous and Spatially Inhomogeneous F-Divergences

In this paper we investigate the construction and the properties of spatially inhomogeneous divergences, functionals arising from optimal Entropy-Transport problems that are computed in terms of an entropy function $F$ and a cost function. Starting from the power-like entropy $F(s)=(s{p}-p(s-1)-1)/ (p(p-1))$ and a suitable cost depending on a metric $\mathsf {d}$ on a space $X$ , our main result ensures that for every $p>1$ the related inhomogeneous divergence induces a distance on the space of finite measures over $X$. We also study in detail the pure entropic setting, that can be recovered as a particular case when the transport is forbidden. In this situation, corresponding to the classical theory of $F$ -divergences, we show that the construction naturally produces a symmetric divergence and we highlight the important role played by the class of Matusita's divergences.