Let us say that two metrics $d$ and $d_0$ on a set $X$ are *related* if there exist positive constants $0 < \alpha \leq \beta$ such that
$$
\alpha \,\left(d_0(x,y) + d_0(y,z) - d_0(x,z)\right) \leq
d(x,y) + d(y,z) - d(x,z) \leq \beta \, \left(d_0(x,y) + d_0(y,z) - d_0(x,z)\right)
$$
for all $x$, $y$, and $z$ in $X$.

A simple and key remark is the following

**Proposition.** *Two related metrics are bi-lipschitz and have exactly the same geodesics.*

By having the same geodesics I mean that three points $x,y,z \in X$ satisfy $$ d(x,y) + d(y,z) - d(x,z) = 0 $$ if and only if they satisfy $$ d_0(x,y) + d_0(y,z) - d_0(x,z)=0. $$

Geometrically speaking, two metrics are related if the defects in the triangle inequality for one of the distances are uniformly controlled by the defects of the other distance.

Question.Is there a simple way to describe all metrics on $\mathbb{R}^3$ that are related to the $\ell_\infty$ metric $d_\infty(x,y) = \max\{|x_i - y_i| : 1 \leq i \leq 3\}$?

In a previous MO question I was exploring "related" norms, which I called "strongly equivalent" and I had not seen the relation with the work of Tony Thompson on the geometry of cones. This is explained in what follows.

**Motivation and explanation of the title.**

A different insight on related metrics is gained if we think of the cone $\mathcal{M}(X)$ of all semi-metrics (i.e., $d(x,y)$ could be zero for $x \neq y$) on the set $X$.

This cone (as every other cone) defines a partial order: $d_1 \preceq d_2$ if $d_2 - d_1$ is a semi-metric. Note that two metrics $d$ and $d_0$ are related if and only if there exist constants $0 < \alpha \leq \beta$ such that
$$
d \preceq \beta d_0 \; \hbox{and} \; \alpha d_0 \preceq d .
$$
In other words two metrics are related if and only if they are in the same *Thompson component* of the metric cone $\mathcal{M}(X)$.

Hilbert's fourth problem is *basically* :

*Describe the Thompson component of the Euclidean metric in $\mathcal{M}(\mathbb{R}^n)$.*

The answer to this problem in the plane is

**Theorem.** *A metric $d$ on the plane is related to the Euclidean metric if and only if there exists a Lebesgue-measurable function $f$ on the space of straight lines such that
$$
0 < \epsilon \leq f(\ell) \leq K < \infty
$$
for some positive constants $\epsilon, K$ and for almost all lines $\ell$ such that the distance between any two points $x, y \in \mathbb{R}^2$ is given by the
integral of $f$ over the set of all lines that separate them.*

This follows easily from the work of Alexander and Ambarzumian (and Busemann and Pogorelov) on Hilbert's fourth problem.

A neat variation is to describe the Thompson component of the $\ell_1$ metric on $\mathbb{R}^n$.

**Theorem.** *A metric $d$ on $\mathbb{R}^n$ is related to the $\ell_1$ (taxicab) metric if and only if
$$
d(x,y) = \sum_{i=1}^n |f_i(x_i) - f_i(y_i)|,
$$
where the $f_i : \mathbb{R} \rightarrow \mathbb{R}$ $(1 \leq i \leq n)$ are
strictly increasing functions satifying
$$
0 < \epsilon \leq \frac{f_i(t)-f_i(s)}{t-s} \leq K < \infty
$$
for some positive constants $\epsilon, K$ and all real numbers $t \neq s$.*

This result can be generalized to other polytopal norms so long as they are hypermetric (they can be isometrically embedded into an $\ell_1$ space), but $\ell_\infty$ is not like that and seems a bit of a challenge. Thus the OP:

Describe the Thompson component of the $\ell_\infty$ metric in $\mathcal{M}(\mathbb{R}^3)$.