  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.
