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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 $\mathcal{M}(\mathbb{R}^3)$.

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