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In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...

We consider the f-divergence, which takes the form $$ D_f(P \| Q) = \int_\Omega f\left(\frac{dP}{dQ}\right) dQ. $$ For example, when $f(t) = t \log t$, we obtain the KL-divergence. My question is ...