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 to in France, the most efficient solution is to reduce the problem to going from to Paris and then from Paris to .
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 be a point in a metric space . Then the SNCF metric with respect to is defined by
It is easy to see that is a metric.
Now, what if the train from to Paris stops over in 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 “ lies on the straight line defined by and ” is required, so the definition becomes more specialised:
Definition 2 (SNCF metric, enhanced version).
Let be the origin in the space with Euclidean norm . Then the SNCF metric is defined by
The metric space is, in addition, a real tree since if and do not lie on the same http://planetmath.org/node/6962ray from , the only arc in joining and consists of the two ray http://planetmath.org/node/5783segments and . Other injections which are arcs in Euclidean do not remain continuous in .
|Date of creation||2013-03-22 15:17:25|
|Last modified on||2013-03-22 15:17:25|
|Last modified by||GrafZahl (9234)|