Correspondence between a maximum ratio and a metric Define
$$P=\left \{ x \in \mathbb{R}^n : x_i \geq 0,\sum_{i=1}^n x_i=1 \right \} \\
X=\{ (p,q) \in P \times P : q_i=0 \Rightarrow p_i=0 \}.$$
Or in words: $P$ is the set of all probability distributions on some set with $n$ elements, and $X$ is the set of pairs $(p,q)$ of such distributions such that $p$ is absolutely continuous with respect to $q$. 
I'm looking at a function $f : X \to \mathbb{R},f((p,q))=\max_i \frac{p_i}{q_i}$, with the understanding $0/0=1$. I'd like to think of this as a kind of "distance" between the distributions $p$ and $q$ which bottoms out at $1$ when $p=q$. Note that $p=q$ is indeed the strict minimum of $f((\cdot,q))$.
Accordingly, is it possible to obtain a metric $d$ such that $\lim_{k \to \infty} f(p^{(k)},q)=1$ if and only if $\lim_{k \to \infty} d(p^{(k)},q)=0$? 
It would be nice if such a $d$ did not depend explicitly on $n$. It would also be of interest if $d$ were merely "metric-like"; this is a soft concept, but for example the Kullback-Leibler divergence is "metric-like" in this sense.
A concrete idea that comes to mind is $d(p,q)=\| \log(p/q) \|_\infty$, which is of course a bona fide metric. However this is not necessarily just $\log f$, because it could be that $\max |\log(p/q)| = | \min \log(p/q)|$. Can that happen? If so does it break the desired convergence property? Finally, it would be really nice if these two had the same monotonicity structure, i.e. $f((p_1,q_1)) \leq f((p_2,q_2))$ if and only if $d(p_1,q_1) \leq d(p_2,q_2)$, though that may be too much to hope for.
 A: The function $f$ as well as the metric $d$ mentioned in OP are closely related to the geometry of cones.
More precisely, the Funk (weak) metric on the convex cone $\Bbb R^n_{++}=\{x\mid x_i>0,\forall i\}$ is defined as
$$\operatorname{Funk}(x,y) = \ln\Big(\max_{i=1,\ldots,n}\frac{x_i}{y_i}\Big).$$
In particular, $\operatorname{Funk}(x,y)=\ln(f(x,y))$. This definition can be extended to the whole nonnegative orthant (i.e. the closure of $\Bbb R_{++}^n$) by allowing $\operatorname{Funk}(x,y)$ to have infinite values. We refer to Chapter 2 of the Handbook on Hilbert Geometry by Papadopoulos and Troyanov for the technical details.
Now, note that 
$$d_T(x,y)=\|\ln(x)-\ln(y)\|_{\infty} = \max\big\{\operatorname{Funk}(x,y),\operatorname{Funk}(y,x)\big\}$$ is the so-called Thompson metric on $\Bbb R^n_{++}$. Furthermore, you might be interested by their projective counterpart which is defined as $$d_H(x,y) = \operatorname{Funk}(x,y)+\operatorname{Funk}(y,x)$$ and is called the Hilbert projective metric. For a smooth introduction to these metrics I strongly recommend the book on Nonlinear Perron-Frobenius theory by Lemmens and Nussbaum (Chapter 2 again).
