Let $X, Y$ be random variables with densites $X = f_x dx$ and $Y = f_ydx$ with respect to the Lesbegue measure. I'm interested in the metric
$$d(X, Y)^2 = \frac{1}{2}\int \frac{(f_x - f_y)^2}{f_x+f_y}dx$$
This is called the Le Cam metric, or triangular discrimination (technically they differ by a constant factor). I had a previous question about it here.
An alternative way to prove that it is a metric is done via this paper. I have reproduced the argument below
By Berg, Christensen, and Ressel [4, Ch. 3 (Proposition 3.2)] it suffices to show that the kernel is negative-definite. So let there be given two finite sequences of real numbers, $(c_i)$ and $(p_i)$, and assume that $\sum_i c_i = 0$ and that all the $p_i$ are positive. Then $$ \sum_{i,j}\frac{(p_i-p_j)^2}{p_i+p_j}c_ic_j = -4\sum_{i,j}\frac{p_ip_j}{p_i+p_j}c_ic_j =-4\int_0^\infty\left(\sum_i c_i p_i \exp(-tp_i)\right)^2dt\geq 0. $$
The citation is to Harmonic Analysis on Semigroups. My understanding is that this implies that there is some map $F : \mathbb{R}_{\geq 0}^n\to \mathcal{H}$ into a hilbert space such that
$$d(X,Y) = \lVert F(X) - F(Y)\rVert_\mathcal{H}$$
I can almost see this directly. In particular, one can write
$$d(X,Y) = \lVert \frac{f_x + f_y}{2} - \frac{1}{(1/f_x) + (1/f_y)} \rVert_2$$
E.g. $d$ is the $\ell_2$-norm of the difference between the arithmetic mean and harmonic mean. But I do not see the existence of the map $F$ that I am interested in yet.
Does such an $F$ have an explicit form?