Homogeneous but non-translation invariant metrics on $\mathbb{R}^2$ Besides the various "railway" metrics (British, French, etc.), are there any other common examples of metrics on $\mathbb{R}^2$ that are both homogeneous AND non-translation invariant?
 A: If you apply any non-linear homeomorphism of $\Bbb R^2$ to the standard metric, you'll get a homogeneous non-translation invariant metric. For example
$$d(x_1,x_2,y_1,y_2)=\sqrt{(x_1-x_2)^2+(e^{y_1}-e^{y_2})^2}$$
In general this takes the form $d(x,y)=d'(f(x),f(y))$, where $f$ is a nonlinear homeomorphism and $d'$ is the standard metric or any other homogeneous metric. This does not exhaust all possibilities but gives a wide variety of metrics.

Edit: As the OP has clarified, the example above uses the wrong notion of "homogeneous". We instead want a non-translation invariant metric satisfying $d(\alpha x,\alpha y)=\alpha d(x,y)$ for all $\alpha\in\Bbb R^{\ge0}$, where $\alpha x$ is scalar multiplication.
This condition isn't very translation invariant to begin with, as it clearly treats $(0,0)$ specially, so one can expect there to be many such metrics. Since it simplifies the coordinates, let's assume that points in $\Bbb R^2$ are identified with polar coordinate space $\Bbb R^{\ge0}\times\Theta$ (where $\Theta=(-\pi,\pi]$).
Pick any metric $d_\theta$ on $\Theta$, and take $$d(r_1,\theta_1,r_2,\theta_2)=\inf_{r_3\in\Bbb R^{\ge0}}(|r_1-r_3|+|r_2-r_3|+r_3d_\theta(\theta_1,\theta_2)).$$
This can be understood as a kind of railway metric where one is forced to travel radially from $r_1$ to some $r_3$ at a cost of $|r_1-r_3|$, then laterally along the circle, with distance function $r_3d_\theta$, and finally radially from $r_3$ to $r_2$. Verifying that this satisfies the triangle inequality is not difficult and left as an exercise. The infimum is also explicitly calculable, although it depends on the value of $d_\theta(\theta_1,\theta_2)$. If $d_\theta(\theta_1,\theta_2)\ge2$, the best path is always the one that goes through the center, so we get the spoke-like metric $d(r_1,\theta_1,r_2,\theta_2)=|r_1|+|r_2|$. For $d_\theta(\theta_1,\theta_2)\le2$ the best path takes $r_3=\min(r_1,r_2)$, so we have $d(r_1,\theta_1,r_2,\theta_2)=|r_1-r_2|+\min(r_1,r_2)d_\theta(\theta_1,\theta_2))$.
It should be clear that with $d_\theta=2$  the resulting metric is not translation invariant; if we consider the generated topology (which is homeomorphic to the open cone on an uncountable discrete space, using a half-open interval $[0,1)$ in the construction instead of $[0,1]$) removing $0$ disconnects the space into uncountably many copies of $\Bbb R$, but removing any other point only disconnects the space into two pieces.
