Math Is Fun Forum

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Can the following be a ball from a given metric?

The question is as follows:

Is it possible to find a metric in R^2 such that the ball of radius 1 centered at 0 looks like:

[-10, 10] x [-1/10, 1/10]  U  [-1/10, 1/10] x [-10, 10]

At first I thought it was possible and scratched my head many hours and after writing down a lot of equations and verifying I came up with the following "metric":

d((x1,y1), (x2,y2)) = min { max { 10|x1-x2|, |y1-y2|/10 }, max { |x1-x2|/10, 10|y1-y2| } }

Surprisingly this DOES give the above picture as the ball, however, triangular inequality fails, and hence its not a metric.

Turns out that it is not possible to find a metric. Professor hinted "you can use the vector space structure of R^2", but i dont know what to make of it.

Also, now that I think about it. Probably the reason no metric can give such a ball is because the figure is not convex? Although as far as I know, a norm will always give convex balls, while metrics can give both convex and nonconvex?

Any help....? I have suffered enough.

Last edited by LuisRodg (2011-01-26 14:09:11)