I know that KL divergence does not satisfy the triangle inequality, in that
$$ D(q||p) \leq D(q||r) + D(r||p) $$
is not always true. This statement can be readily proven by providing a counterexample that violates the inequality above.
However, is it true that the KL divergence never satisfies the triangle inequality? Or are there some cases where the above inequality holds and some where it does not? I have never seen a general proof that KL divergence never satisfies the triangle inequality, nor have I come across an example where it does satisfy.
Thank you for your insights!