I'm trying to understand an example that occurs very early in Gromov's book "Metric Structures for Riemannian and Non-Riemannian Spaces".
On page 3, he introduces the following metric on $\mathbb{R}^n$. Consider points $x_1,x_2 \in \mathbb{R}^n$. Write them in polar coordinates as $x_i = r_i s_i$, where $r_i \geq 0$ and $s_i \in S^{n-1}$. We then define $$d(x_1,x_2) = |r_1-r_2| + \text{min}(r_1,r_2) \cdot \|s_1-s_2\|^{1/2},$$ where $\|\cdot\|$ is the usual Euclidean distance on the sphere. You can then define an alternate metric $d_{\ell}(x_1,x_2)$ to equal the infimum of the lengths of all continuous paths from $x_1$ to $x_2$, where the length of a path is measured using $d(\cdot,\cdot)$ in the usual way.
What he claims is that $$d_{\ell}(x_1,x_2) = \begin{cases} |r_1-r_2| & \text{if $s_1 = s_2$},\\ r_1+r_2 & \text{if $s_1 \neq s_2$}. \end{cases}$$ In particular, $d_{\ell}$ induces a non-standard topology on $\mathbb{R}^n$ where all spheres centered at the origin are discrete.
I'm having trouble proving the above formula for $d_{\ell}$. What must be the case is that if $s_1 \neq s_2$ then paths that move the $S^{n-1}$-coordinate when the radius is nonzero must have infinite length, and this must somehow come from the $1/2$ in the exponent of $\|s_1-s_2\|$. Can anyone help me?
