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

[radou]

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.

 

[radou]

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 ?