What is the significance of strong equivalence in topological space metrics?

[ForMyThunder]

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

d_p((x,y),(x₁,y₁))=((d_X(x,y))^p+(d_Y(x₁,y₁))^p)^1/p

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

as p\rightarrow\infty,

d_\infty=max((d_X(x,y),d_Y(x₁,y₁)).

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

Each of the d_p((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!