SNCF metric

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

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

Definition 1 (SNCF metric).

Let $P$ be a point in a metric space $(F,d)$ . Then the SNCFmetric $d_{P}$ 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 theride (or the other way round)? Sure, Paris is a beautiful city, butyou wouldn’t always want to go there and back again. Toimplement this, the geometric notion of “ $y$ lies on the straightline defined by $x$ and $P$ ” is required, so the definition becomesmore 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 sinceif $x$ 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 $x P$ and $y P$ . Other injections which are arcs in Euclidean $\\mathbb{R}^{n}$ do not remain continuous in $(\\mathbb{R}^{n},d_{P})$ .