Are all Distance Metrics, Divergence Functions?

Question

Quite simply, as the question states. Are all distance metrics, divergence functions? The wikipedia defintion for a divergence function is given as

Suppose S is a space of all probability distributions with common support. Then a divergence on S is a function D(· || ·): S×S → R satisfying

D(p || q) ≥ 0 for all p, q ∈ S, D(p || q) = 0 if and only if p = q,

https://en.wikipedia.org/wiki/Divergence_(statistics)

whilst the distance metric definition (https://en.wikipedia.org/wiki/Metric_(mathematics)) given is similar with the addition of requiring symmetry and abiding of the triangle inequality.

Thanks for your help.

Answer 1

Yes.

If $d$ is a distance metric on $S$ then $d$ is a divergence on $S$.

Answer 2

Yes, a metric on a space of probability distributions with common support is, a fortiori, a divergence.