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Suppose you have a list of $n$ dimensional vectors. One can order them by using an $L_p$ norm to do comparisons between vectors. The general questions is, will the order be different depending on the choice of $p$?

I am mainly interested in $p=1,2$.

For example, suppose I have $x_1, x_2, x_3$ are $d$-dim vectors ($d>1$), and

$L_1(x_1) < L_1(x_2) < L_1(x_3) $, does this guarantee $L_2(x_1) < L_2(x_2) < L_2(x_3) $ also?

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Can you explain in more detail how your ordering procedure works? – Nate Eldredge Dec 2 '12 at 20:46
it's just like sorting any array. It doesn't matter what the algo to sort is, the order of the vectors is dependent on the comparison operator. – Cam.Davidson.Pilon Dec 2 '12 at 22:17
up vote 1 down vote accepted

Definitely not.

Take $d=2$, $x_1=(1,0)$, $x_2 = (0.6,0.6)$. Then in your notation we have $L_1(x_1) = L_2(x_1) = 1$, while $L_1(x_2) = 1.2$, $L_2(x_2) \approx 0.848$.

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