On $f$-divergences between Cauchy distributions

We prove that the $f$-divergences between univariate Cauchy distributions are always symmetric and can be expressed as functions of the chi-squared divergence. We explicit the corresponding functions for the total variation distance, the Kullback-Leibler divergence, the LeCam-Vincze divergence, the squared Hellinger divergence, and the Jensen-Shannon divergence. We then show that this symmetric $f$-divergence property does not hold anymore for multivariate Cauchy distributions. Finally, we present several metrizations of $f$-divergences between univariate Cauchy distributions.