Metric Spaces-BASIC DISTANCE

1. Sep 8, 2010

mynameisfunk

OK, for metric spaces there are apparantly 3 different possibilities for the distance function in M where M is the usual Euclidean Plane:

(A) D(u,v) = sqrt((x1-x2)2 + (y1-y2)2)
(B) D(u,v) = max(|x1-x2|,|y1-y2|)
(C) D(u,v) = |x1-x2| + |y1-y2|
which somehow correspond to the picture I have attached.
A corresponds to the circle, B to the square and C to the diamond(this is supposed to be a square diamond but i created the image in paint, sorry)
Now, I understand (A) but I cannot seem to understand why (B) and (C) end up looking this way. and to be honest, I dont understand B and C at all.

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Last edited: Sep 8, 2010
2. Sep 8, 2010

trambolin

These are the unit balls with respect to each metric. In other words they mark the points which have the distance "1" to the origin (x_2, y_2) = (0,0). So the first one is a circle equation. The second one has the max of any coordinates, therefore max of (1,1) is 1 which is on the square. So figure out the diamond...

And you have definitely much more choices than 3. These are the most common three.

3. Sep 9, 2010

Landau

To elaborate on trambolin's last sentence: these three are instances of a special class of metrics defined for every real p>=1:

$$d_p(x,y)= \left(|x_1-y_1|^p+|x_2-y_2|^p\right)^{1/p}$$

(A) corresponds to p=2
(C) corresponds to p=1
(B) is the extension for $p=\infty$
Besides these p-metrics there are lots of other metrics on R^2.

Last edited: Sep 9, 2010