Are d1 and dp Metrics Uniformly Equivalent in R^n?

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In summary, the conversation is about proving the uniform equivalence between the metrics d1, dp, and d∞ in R^n. The speaker has shown that d1 ~ d∞ and is now seeking help to show that d1 ~ dp. They suggest comparing the p-th power of ∑|ai - bi| with ∑|ai - bi|^p, but are unsure if (∑|ai - bi|)^p is always greater than ∑|ai - bi|^p.
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radou
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Well, again I'm a bit stuck.

I have to prove that the metrics d1, dp (where p is from <1, ∞>) and d∞ in R^n are uniformly equivalent. The metrics are given with:

d1(a, b) = ∑|ai - bi|
dp(a, b) = (∑|ai - bi|^p)^(1/p)
d∞(a, b) = max{|ai - bi|, i = 1, ... ,n} (of course, the sums are ranging from 1 to n)

The relation of uniform equivalence between metrics is an equivalence relation, so if d1 ~ d∞ and d1 ~ dp, then dp ~ d∞.

I have shown that d1 ~ d∞, but I am stuck with showing that d1 ~ dp, and would be most grateful for a push here.
 
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Perhaps an idea would be to use the fact to compare the p-th power of ∑|ai - bi| with ∑|ai - bi|^p ? I'm not sure if in general (∑|ai - bi|)^p (i.e. a multinomial expansion) is greater than ∑|ai - bi|^p ?
 

FAQ: Are d1 and dp Metrics Uniformly Equivalent in R^n?

What are Uniformly Equivalent Metrics?

Uniformly equivalent metrics are two different ways of measuring distance that result in the same topological structure. This means that they both define the same open and closed sets, and preserve continuity of functions.

How are Uniformly Equivalent Metrics related to Metric Spaces?

Uniformly equivalent metrics are a concept within metric spaces. A metric space is a set of points with a defined distance function, and uniformly equivalent metrics are two different distance functions that result in the same metric space.

What is the importance of Uniformly Equivalent Metrics?

Uniformly equivalent metrics are important because they allow us to have different ways of measuring distance while still preserving the same topological structure. This can be useful in various mathematical and scientific fields, such as optimization and analysis.

How can Uniformly Equivalent Metrics be identified?

Uniformly equivalent metrics can be identified by showing that the two metrics satisfy certain conditions, such as being bounded by each other and having the same convergence sequences. If these conditions are met, then the metrics are uniformly equivalent.

Can Uniformly Equivalent Metrics have different properties?

Yes, uniformly equivalent metrics can have different properties. While they both define the same topological structure, they may have different properties such as different convergence rates or different boundedness. However, they will still result in the same metric space.

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