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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<sup>p</sup> - p(s -1) - 1)/(p(p-1)) and a suitable cost depending on a metric d on a space X, our main result ensures that for every p &gt; 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.

Metric Properties of Homogeneous and Spatially Inhomogeneous $F$ -Divergences