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?