*preprint*

**Inserted:** 19 feb 2019

**Year:** 2019

**Abstract:**

In this paper I investigate the construction and the properties of the so-called marginal perspective cost $H$, a function related to Optimal Entropy-Transport problems obtained by a minimizing procedure, involving a cost function $c$ and an entropy function. In the pure entropic case, which corresponds to the choice $c=0$, the function $H$ naturally produces a symmetric divergence. I consider various examples of entropies and I compute the induced marginal perspective function, which includes some well-known functionals like the Hellinger distance, the Jensen-Shannon divergence and the Kullback-Liebler divergence. I discuss the metric properties of these functions and I highlight the important role of the so-called Matusita divergences. In the entropy-transport case, starting from the power like entropy $F_p(s)=(s^p-p(s-1)-1)/(p(p-1))$ and the cost $c=d^2$ for a given metric $d$, the main result of the paper ensures that for every $p>1$ the induced marginal perspective cost $H_p$ is the square of a metric on the corresponding cone space.