Postby **Mathemagician** » Fri Oct 22, 2010 12:41 pm

Let D represent the candidate new metric:

A) D = 4+d

d(x, x) = 0 => D(x, x) = 4, so D is clearly not a metric as a point has a distance from itself.

B) e^d -1

Let d be the regular Euclidean metric and consider x = (0, 0), y = (0, 1), and z = (0, 2). It should be clear that going directly from x to z is "longer" than going from x to y to z, violating the triangle inequality.

C) d - |d|

Suppose x and y are distinct. Then d(x, y) = m > 0. But D(x, y) = m - m = 0, so two distinct points have no distance separating them, so D is not a metric.

D) d^2

Consider the same environment as in B. Going directly from 0 to 2 is a distance of 4, whereas going in two discrete steps is a distance of 2.

E) sqrt (d)

Fairly easy to convince yourself that this is reasonable and the triangle inequality probably holds by considering the graph.