Metrics

[ForMyThunder]

A metric dp on the topological space X×Y, with dX(x,y) and dY(x1,y1) being metrics on X and Y respectively, is defined as

dp((x,y),(x1,y1))=((dX(x,y))p+(dY(x1,y1))p)1/p

What does each dp((x,y),(x1,y1)) mean (geometrically or visually)?

as p\rightarrow\infty,

d\infty=max((dX(x,y),dY(x1,y1)).

What is the meaning of "max(A,B)"?

Each of the dp((x,y),(x1,y1)) are strongly equivalent; what does this mean geometically?

[morphism]

You probably want p>=1 for d_p to actually be a metric. Anyway, have you tried putting X=Y=R? This should be revealing, e.g. if p=1 you get the "taxicab metric" and if p=2 you get the usual Euclidean metric. For other p>1, you may want to sketch the unit balls of (R^2, d_p) to get a feel for what the metric d_p does.

max(A,B) means what you would expect it to be; namely, max(A,B) is A if A>=B and B otherwise.

Finally, two metrics d and d* on a metric space Z are said to be strongly equivalent if you can find positive constants M and m such that the string of inequalities
m d(x,y) < d*(x,y) < M d(x,y)
holds for all x and y in Z. This essentially means that inside each d-ball you can find a d*-ball and vice versa. In particular, this implies that the metrics d and d* generate the same topology on Z.

[ForMyThunder]

Thanks. I've been worrying about what they were fr a while. It seems kinda obvious now.

Thanks again!