# SNCF metric

The following two examples of a metric space (one of which is a real tree) obtained their name from the of the French railway system. Especially malicious rumour has it that if you want to go by train from $x$ to $y$ in France, the most efficient solution is to reduce the problem to going from $x$ to Paris and then from Paris to $y$.

Since their discovery, the intrinsic laws of the French way of going by train have made it around the world and reached the late-afternoon tutorials of first-term mathematics courses in an effort to lighten the moods in the guise of the following definition:

###### Definition 1 (SNCF metric).

Let $P$ be a point in a metric space $(F,d)$. Then the SNCF metric $d_{P}$ with respect to $P$ is defined by

 $d_{P}(x,y):=\begin{cases}0&\text{if }x=y\\ d(x,P)+d(P,y)&\text{otherwise.}\end{cases}$

It is easy to see that $d_{P}$ is a metric.

Now, what if the train from $x$ to Paris stops over in $y$ during the ride (or the other way round)? Sure, Paris is a beautiful city, but you wouldn’t always want to go there and back again. To implement this, the geometric notion of “$y$ lies on the straight line defined by $x$ and $P$” is required, so the definition becomes more specialised:

###### Definition 2 (SNCF metric, enhanced version).

Let $P$ be the origin in the space $\mathbb{R}^{n}$ with Euclidean norm $\|\cdot\|_{2}$. Then the SNCF metric $d_{P}$ is defined by

 $d_{P}(x,y):=\begin{cases}\|x-y\|_{2}&\text{if }x\text{ and }y\text{ lie on the% same ray from the origin}\\ \|x\|_{2}+\|y\|_{2}&\text{otherwise}\end{cases}.$

The metric space $(\mathbb{R}^{n},d_{P})$ is, in addition, a real tree since if $x$ and $y$ do not lie on the same http://planetmath.org/node/6962ray from $P$, the only arc in $(\mathbb{R}^{n},d_{P})$ joining $x$ and $y$ consists of the two ray http://planetmath.org/node/5783segments $xP$ and $yP$. Other injections which are arcs in Euclidean $\mathbb{R}^{n}$ do not remain continuous in $(\mathbb{R}^{n},d_{P})$.

Title SNCF metric SNCFMetric 2013-03-22 15:17:25 2013-03-22 15:17:25 GrafZahl (9234) GrafZahl (9234) 5 GrafZahl (9234) Example msc 51M05 msc 97A20 msc 54E35 RealTree